Constants

List of constants which correspond to the sum of a divergent series and/or to the value of an infinite product and/or to the value of a divergent integral.

$\Gamma(z)$ denotes the gamma function.
$\Gamma'(z)$ denotes the derivative of the gamma function.
$\ln\Gamma(z)$ denotes the log-gamma function.
$\Psi^{(0)}(z) = \Psi(z) = \frac{d}{dz} \ln\Gamma(z) = \frac{\Gamma'(z)}{\Gamma(z)}$ denotes the digamma function.
$\Psi^{(1)}(z) = \frac{d}{dz} \Psi^{(0)}(z)$ denotes the trigamma function.
$\zeta(z)$ denotes the Riemann zeta function.
$\zeta'(z)$ denotes the derivative of the Riemann zeta function.
$\zeta(z,a)$ denotes the Hurwitz zeta function.
$\zeta'(z,a)$ denotes the derivative of the Hurwitz zeta function.
$\ln Z(z)$ denotes the log-zeta function.
$Z^{(0)}(z) = Z(z) = \frac{d}{dz} \ln\zeta(z) = \frac{\zeta'(z)}{\zeta(z)}$ denotes the dizeta function.
$P(z)$ denotes the prime zeta function.
$P'(z)$ denotes the derivative of the prime zeta function.
$\xi(z)$ denotes the Riemann xi function.
$\xi'(z)$ denotes the derivative of the Riemann xi function.
$\eta(z)$ denotes the Dirichlet eta function.
$\eta'(z)$ denotes the derivative of the Dirichlet eta function.
$K(z)$ denotes the K-function.
$K'(z)$ denotes the derivative of the K-function.
$\text{Ei}(z)$ denotes the exponential integral function.
$\text{Li}_s(z)$ denotes the polylogarithm function (also known as Jonquière’s function).
$\theta_3(z,q)$ denotes the Jacobi theta_3 function.
$\theta_3'(z,q)$ denotes the derivative of the Jacobi theta_3 function.

$\zeta(1) = \gamma \approx 0.57721566490153286060651209008240243\ldots$ denotes the Euler-Mascheroni constant.
$\zeta(2) = \dfrac{\pi^2}{6} \approx 1.6449340668482264364724151666460251\ldots$ denotes the Basel constant.
$\zeta(3) \approx 1.202056903159594285399738161511449990764\ldots$ denotes the Apéry constant.
$\varpi \approx 2.62205755429211981046483958989111941368275\ldots$ denotes the lemniscate constant.
$A = A_1 \approx 1.282427129100622636875342568869791727\ldots$ denotes the Glaisher-Kinkelin constant.
$C = \beta(2) \approx 0.91596559417721901505460351493238411\ldots$ denotes the Catalan constant.
$G \approx 0.59634736232319407434107849936927937607417\ldots$ denotes the Euler-Gompertz constant.

$A_n \; (n \geq 0)$ denotes the Glaisher-Kinkelin constants (with $A_0 = \sqrt{2\pi} \;$ and $A_1 = A$ ).
$B_n \; (n \geq 0)$ denotes the Bernoulli numbers (with $B_0 = 0$ and $B_1 = \pm \frac{1}{2}$ ).
$H_n \; (n \geq 0)$ denotes the harmonic numbers (with $H_0 = 0$ ).
$H_{n,r} = \displaystyle \sum_{m=1}^{n} \dfrac{1}{m^r}$ denotes the n-th generalized harmonic number or order r (with $H_{0,r} = 0$ ).
$H_n^{(r)}$ denotes the n-th hyperharmonic number of order r (with $H_0^{(r)} = 0$ ).
$M_n \; (n \geq 0)$ denotes the Meissel-Mertens constants (with $M_0 = \frac{1}{4} + \gamma$ ).
$P_n \; (n \geq 0)$ denotes the prime zeta constants (with $P_0 = -\frac{1}{2}$ ).
$\gamma_n \; (n \geq 0)$ denotes the Stieltjes constants (with $\gamma_0 = \gamma$ ).
$\lambda_n \; (n \geq 0)$ denotes the Li constants (with $\lambda_0 = 0$ ).

$(2\pi)^8 \, \dfrac{\zeta(4)}{\zeta(8)} = 26880\pi^4 \approx 2618356.366993985512915516142672393390392677502\ldots$

$(2\pi)^8 = 256\pi^8 \approx 2429063.9401140669458249153290001320439544779447401171\ldots$

$\dfrac{(2\pi)^8}{\zeta(4)} = 23040\pi^2 \approx 2244305.45742341615392758526514776576319372357389867\ldots$

$(2\pi)^6 \, \dfrac{\zeta(3)}{\zeta(6)} = 60480\,\zeta(3) \approx 72700.4015030922623809761640082124954414663709\ldots$

$(2\pi)^6 = 64\pi^6 \approx 61528.9083888194839699340443937548735275019217971869834\ldots$

$\dfrac{(2\pi)^6}{\zeta(3)} = \dfrac{64\pi^6}{\zeta(3)} \approx 51186.352515501867733131294157792057290161841706706509\ldots$

$(2\pi)^4 \, \dfrac{\zeta(2)}{\zeta(4)} = 240\pi^2 \approx 2368.705056261446068520277839970276272475287857737\ldots$

$(2\pi)^4 = 16\pi^4 \approx 1558.54545654403899578304532301928177999564137076296674\ldots$

$\dfrac{(2\pi)^4}{\zeta(2)} = 24^2\,\zeta(2) = 96 \pi^2 \approx 947.482022504578427408111135988110508990115143\ldots$

$e^{2\pi} \approx 535.491655524764736503049329589047181477805797603294915507205255\ldots$

$\text{Li}_{-5}(\frac{1}{e}) = \dfrac{e(e^4\!+\!26e^3\!+\!66e^2\!+\!26e\!+\!1)}{(e\!-\!1)^6} \\ \\ \approx 119.99782676761602472146261194878073153484833779390673063109535645\ldots$

$2(\cosh(\pi\sqrt{2}) \,-\, \cos(\pi\sqrt{2})) \approx 85.56396788782143768141788282268857675205573\ldots$

$2\pi \sinh(\pi) \approx 72.562869445968505834508589102080777748781358021830402883\ldots$

$\dfrac{24\,\zeta(2)}{\zeta(1)} = \dfrac{4\pi^2}{\gamma} \approx 68.394570703641683395011835169862452214443485037528049\ldots$

$\cosh(\pi\sqrt{2}) \,-\, \cos(\pi\sqrt{2}) \approx 42.78198394391071884070894141134428837602786833\ldots$

$24\,\zeta(2) = (2\pi)^2 = 4\pi^2 \approx 39.47841760435743447533796399950460454125479762\ldots$

$2\sqrt{2\pi} \, \cosh\left(\dfrac{\pi\sqrt{3}}{2}\right) \approx 38.2430420997420228014238002456335439947654844948\ldots$

$\pi \sinh(\pi) \approx 36.28143472298425291725429455104038887439067901091520144174\ldots$

$\dfrac{\zeta'(-1)}{\zeta(-1)} - \dfrac{\zeta'(2)}{\zeta(2)} = -1 - \gamma - \ln(2\pi) + 24 \ln A \\ \\ \approx 29.3631583671040649408410500899813637556595799725845381850896902059\ldots$

$\text{Li}_{-4}(\frac{1}{e}) = \dfrac{e(e^3\!+\!11e^2\!+\!11e\!+\!1)}{(e\!-\!1)^5} \\ \\ \approx 24.0033329747690522721292121894483921207140518043451568930078000598\ldots$

$e^\pi \approx 23.1406926327792690057290863679485473802661062426002119934450464\ldots$

$2\sinh(\pi) \approx 23.09747871451549675595466863077681936899037813278957891046\ldots$

$\sqrt{2\pi} \, \cosh\left(\dfrac{\pi\sqrt{3}}{2}\right) \approx 19.12152104987101140071190012281677199738274224740\ldots$

$(2\pi)^\frac{3}{2} \approx 15.74960994572241974429064599446720454231168538279873385825730\ldots$

$(2\pi)^2 \, \dfrac{\zeta(1)}{\zeta(2)} = 24\,\zeta(1) = 24\gamma \approx 13.8531759576367886545562901619776583450118\ldots$

$\dfrac{2\sqrt{2\pi}}{3} \, \cosh\left(\dfrac{\pi\sqrt{3}}{2}\right) \approx 12.7476806999140076004746000818778479982551614982\ldots$

$\dfrac{\zeta'(-1)}{\zeta(-1)} - \dfrac{\zeta'(0)}{\zeta(0)} = -1 - \ln(2\pi) + 12 \ln A \\ \\ \approx 12.5512484827981261589434513536262654534894721806244474791305771799\ldots$

$\Gamma(\frac{9}{2}) = \dfrac{105\sqrt{\pi}}{16} \approx 11.631728396567448929144224109426265262108918305803165\ldots$

$\sinh(\pi) \approx 11.548739357257748377977334315388409684495189066394789455232\ldots$

$\cot\!\left(\dfrac{\pi}{32}\right) = \dfrac{\sqrt{2+\sqrt{2\!+\!\sqrt{2+\sqrt{2}}}}}{\sqrt{2-\sqrt{2\!+\!\sqrt{2+\sqrt{2}}}}}\\ \\ \approx 10.153170387608860462107147663419472203767440954850176952529869064\ldots$

$\cot\!\left(\dfrac{\pi}{30}\right) = \sqrt{23+10\sqrt{5}+2\sqrt{3(85+38\sqrt{5})}} \\ \\ \approx 9.5143644542225849296839714549456824666487681451506592273112648914\ldots$

$2\pi\sqrt{2} \approx 8.88576587631673249403176198012138739722924337875138044617079\ldots$

$\pi\,e \approx 8.53973422267356706546355086954657449503488853576511496187960113\ldots$

$\text{Ei}(e) \approx 8.21168165538361560418860406753936903309605065909442196017788\ldots$

$\cot\!\left(\dfrac{\pi}{24}\right) = 2+\sqrt{6}+\sqrt{5\!+\!2\sqrt{6}} \\ \\ \approx 7.5957541127251504405264191404214618374784246098439988296651792846\ldots$

$2\,\xi(10) = \dfrac{16\pi^5}{693} \approx 7.065389547712125905056076422359263641680824395453390\ldots$

$\cot\!\left(\dfrac{\pi}{20}\right) = 1+\sqrt{5}+\sqrt{5\!+\!2\sqrt{5}} \\ \\ \approx 6.3137515146750430989794642447681860594473205031493181513100534957\ldots$

$-\dfrac{\xi(-1)}{\zeta(-1)} = 2\pi \approx 6.2831853071795864769252867665590057683943387987502116\ldots$

$\text{Li}_{-3}(\frac{1}{e}) = \dfrac{e(e^2\!+\!4e\!+\!1)}{(e\!-\!1)^4} \\ \\ \approx 6.0065127966367601482732973028999783503057010217124047825938339232\ldots$

$\cot\!\left(\dfrac{\pi}{19}\right) \approx 5.99267145852349326139874856524395835664960232114035141408\ldots$

$\cot\!\left(\dfrac{\pi}{18}\right) \approx 5.67128181961770953099441843986396442162537826068975030321\ldots$

$\cot\!\left(\dfrac{\pi}{17}\right) \approx 5.34952750550977681939381738210297312025039979927837620243\ldots$

$\cot\!\left(\dfrac{\pi}{16}\right) = 1+\sqrt{2}+\sqrt{2(2\!+\!\sqrt{2})} = \dfrac{\sqrt{2+\sqrt{2\!+\!\sqrt{2}}}}{\sqrt{2-\sqrt{2\!+\!\sqrt{2}}}}\\ \\ \approx 5.0273394921258481045149750710640723857371942520754871282744764718\ldots$

$2\sqrt{2\pi} \approx 5.01325654926200100483153056962209050601397348121987663325984\ldots$

$\text{Ei}(2) \approx 4.954234356001890163379505130227035275518053562420042054527095\ldots$

$\dfrac{\zeta(2)}{\zeta(1)^2} = \dfrac{\pi^2}{6\gamma^2} \approx 4.937104019525433330855495923274505254366646460315057006\ldots$

$\dfrac{8\pi}{375A} \, 2^{-\frac{31}{12}} \, e^\frac{19}{3} \approx 4.909607704735092976788964199132508863389191235799245\ldots$

$2\,\xi(9) = \dfrac{945}{2\pi^4}\,\zeta(9) \approx 4.8604186794542386728716160489422594736287077312616\ldots$

$\cot\!\left(\dfrac{\pi}{15}\right) = \sqrt{7+2\sqrt{5}+2\sqrt{3(5+2\sqrt{5})}} \\ \\ \approx 4.7046301094784542335862345374029002756992607478024861722163016616\ldots$

$\delta \approx 4.669201609102990671853203820466201617258185577475768632745651343\ldots$

$\dfrac{\pi}{A} \, 2^\frac{5}{12} \, e^\frac{1}{3} \approx 4.563637794258763375378366146186703274221958782712821962501\ldots$

$\pi\sqrt{2} \approx 4.442882938158366247015880990060693698614621689375690223085395\ldots$

$\dfrac{\zeta'(\frac{1}{2})}{\zeta(\frac{1}{2})} \,+\, \dfrac{\zeta'(-\frac{1}{2})}{\zeta(-\frac{1}{2})} \approx 4.4219178310369730543088364659343855203646829212802494\ldots$

$\cot\!\left(\dfrac{\pi}{14}\right) \approx 4.381286267534823072404689085032695444150222094232289686146\ldots$

$\dfrac{\zeta'(-2)}{\zeta(-2)} = \ln(2\pi) - \gamma + 12 \ln A_1 \\ \\ \approx 4.24571512591322377352118330564220056184907217535806463082680352237\ldots$

$\sqrt{2\pi\,e} \approx 4.13273135412249293846939188429985264944552191699130845155665\ldots$

$\sqrt{6}\;\dfrac{\Gamma(-\frac{1}{6})}{\Gamma(-\frac{2}{3})} \approx 4.1284295488299119936896931502094857184713977227326014464\ldots$

$\cot\!\left(\dfrac{\pi}{13}\right) \approx 4.057159485638116460694772345740907092840997827538349987802\ldots$

$\dfrac{\zeta'(-4)}{\zeta(-4)} = \ln(2\pi) - \gamma - 120 \ln A_3 \\ \\ \approx 3.73942389777406209361751762131182732949370171374443740729551196621\ldots$

$\cot\!\left(\dfrac{\pi}{12}\right) = 2+\sqrt{3} \approx 3.7320508075688772935274463415058723669428052538103\ldots$

$\dfrac{\zeta'(-6)}{\zeta(-6)} = \ln(2\pi) - \gamma + 252 \ln A_5 \\ \\ \approx 3.68838720185120202794521124156830255935397374205125590705270873343\ldots$

$\dfrac{\zeta'(-8)}{\zeta(-8)} = \ln(2\pi) - \gamma - 240 \ln A_7 \\ \\ \approx 3.67864430112673772350486852046988915896679970117068568749866031194\ldots$

$\dfrac{\zeta'(-10)}{\zeta(-10)} = \ln(2\pi) - \gamma + 132 \ln A_9 \\ \\ \approx 3.67645047326397498755791422231340785218224757106033708771502651693\ldots$

$\dfrac{\sinh(\pi)}{\pi} \approx 3.6760779103749777206956974920282606665071563468276302774780\ldots$

$2\ln(2\pi) \approx 3.6757541328186909671213189456224705594455898945511336512686\ldots$

$\Gamma(\frac{1}{3})\,\Gamma(\frac{2}{3}) = \dfrac{\pi}{\sin(\frac{\pi}{3})} = \dfrac{2\pi}{\sqrt{3}} \approx 3.627598728468435701188156515284311464568132\ldots$

$\Gamma(\frac{1}{4}) \approx 3.625609908221908311930685155867672002995167682880065467433377\ldots$

$-\Gamma(-\frac{1}{2}) = 2\sqrt{\pi} \approx 3.544907701811032054596334966682290365595098912244774\ldots$

$\xi(10) = \xi(-9) = \dfrac{8\pi^5}{693} \approx 3.5326947738560629525280382111796318208404121977\ldots$

$\dfrac{\xi(10)}{\zeta(10)} = \dfrac{1080}{\pi^5} \approx 3.5291847344976563104385095924647103086890613989816775\ldots$

$2\,\xi(8) = \dfrac{8\pi^4}{225} \approx 3.46343434787564221285121182893173728887920304613992610\ldots$

$\cot\!\left(\dfrac{\pi}{11}\right) \approx 3.405687238889250009048130346648848831957299986121828140977\ldots$

$\Gamma(\frac{7}{2}) = \dfrac{15\sqrt{\pi}}{8} \approx 3.3233509704478425511840640312646472177454052302294758\ldots$

$\dfrac{\zeta(2)}{\xi(2)} = \pi \approx 3.141592653589793238462643383279502884197169399375105820974\ldots$

$\cot\!\left(\dfrac{\pi}{10}\right) = \sqrt{5\!+\!2\sqrt{5}} \approx 3.0776835371752534025702905760369098240067021435\ldots$

$\text{Ei}(2) - \text{Ei}(1) \approx 3.059116539645953407912984195895401006500992980687334462\ldots$

$H_\frac{19}{2} = H_{-\frac{21}{2}} = \dfrac{62075752}{14549535} \,-\, 2\ln 2 \approx 2.8802166991992192358791056409321684832\ldots$

$1 \,+\, \cot\!\left(\dfrac{1}{2}\right) = 1 \,+\, \dfrac{\sin 1}{1\!-\!\cos 1} \approx 2.83048772171245191926801943896881662375810\ldots$

$\dfrac{\pi}{3A} \, 2^{-\frac{19}{12}} \, e^\frac{7}{3} \approx 2.8100813064148446215835720274756632296081112685571241109\ldots$

$H_\frac{17}{2} = H_{-\frac{19}{2}} = \dfrac{3186538}{765765} \,-\, 2\ln 2 \approx 2.77495354130448239377384248303743164116\ldots$

$\cot\!\left(\dfrac{\pi}{9}\right) \approx 2.7474774194546222787616640264976727177518725991708258215052\ldots$

$\dfrac{\ln 20}{\ln 3} \approx 2.726833027860842041396094636364162104907103646929810544794200\ldots$

$e \approx 2.718281828459045235360287471352662497757247093699959574966967627\ldots$

$\dfrac{\zeta'(\frac{1}{2})}{\zeta(\frac{1}{2})} = \dfrac{\zeta'(\frac{1}{2})}{\zeta(\frac{1}{2})} - \dfrac{\xi'(\frac{1}{2})}{\xi(\frac{1}{2})} = \dfrac{\pi}{4} + \ln 2 + \dfrac{\ln(2\pi)}{2} + \dfrac{\gamma}{2} \\ \\ \approx 2.6860917096128327911164787487248711445072696258117769215844513155\ldots$

$K_0 \approx 2.68545200106530644530971483548179569382038229399446295305115234\ldots$

$\Gamma(\frac{1}{3}) = \dfrac{2\pi}{\sqrt{3}\,\Gamma(\frac{2}{3})} \approx 2.678938534707747633655692940974677644128689377957301\ldots$

$H_\frac{15}{2} = H_{-\frac{17}{2}} = \dfrac{182144}{45045} \,-\, 2\ln 2 \approx 2.65730648248095298200913660068449046469\ldots$

$\varpi = \dfrac{\left(\Gamma(\frac{1}{4})\right)^2}{2\sqrt{2\pi}} \approx 2.62205755429211981046483958989111941368275495143162316\ldots$

$\zeta(\frac{3}{2}) \approx 2.612375348685488343348567567924071630570800652400063407573328\ldots$

$2\,\xi(7) = \dfrac{315}{4\pi^3}\,\zeta(7) \approx 2.5610139009137739196767306659440617292020964589110\ldots$

$H_\frac{13}{2} = H_{-\frac{15}{2}} = \dfrac{176138}{45045} \,-\, 2\ln 2 \approx 2.52397314914761964867580326735115713135\ldots$

$A_0 = \sqrt{2\pi} \approx 2.5066282746310005024157652848110452530069867406099383166\ldots$

$\alpha \approx 2.502907875095892822283902873218215786381271376727149977336192056\ldots$

$\dfrac{\pi^2}{4} \approx 2.46740110027233965470862274996903778382842485181019765660333734\ldots$

$\xi(9) = \xi(-8) = \dfrac{945}{4\pi^4}\,\zeta(9) \approx 2.4302093397271193364358080244711297368143538\ldots$

$\dfrac{\xi(9)}{\zeta(9)} = \dfrac{945}{4\pi^4} \approx 2.4253383076691741884373450468567914910068181879220395671\ldots$

$\dfrac{\zeta'(1)}{\zeta(1)} + \dfrac{\zeta'(0)}{\zeta(0)} = -H_0 + \ln(2\pi) + \gamma = \ln(2\pi) + \gamma \\ \\ \approx 2.41509273131087834416717156289363771076495428321549042444007031585\ldots$

$\cot\!\left(\dfrac{\pi}{8}\right) = 1+\sqrt{2} = \sqrt{\dfrac{2\!+\!\sqrt{2}}{2\!-\!\sqrt{2}}} \approx 2.414213562373095048801688724209698078569\ldots$

$H_\frac{11}{2} = H_{-\frac{13}{2}} = \dfrac{13016}{3465} \,-\, 2\ln 2 \approx 2.370126995301465802521957113505003285205\ldots$

$\Gamma(-\frac{3}{2}) = \dfrac{4\sqrt{\pi}}{3} \approx 2.3632718012073547030642233111215269103967326081631828\ldots$

$\dfrac{\pi}{A} \, 2^{-\frac{7}{12}} \, e^\frac{1}{3} \approx 2.28181889712938168768918307309335163711097939135641098125\ldots$

$\Psi^{(0)}(10) = \dfrac{\Gamma'(10)}{\Gamma(10)} = \dfrac{\Gamma'(10)}{362880} = H_9 \! - \gamma = \Psi^{(0)}(-9) = \dfrac{\Gamma'(-9)}{\Gamma(-9)} = \\ \\ H_{-10} \,-\, \gamma = \dfrac{7129}{2520} \,-\, \gamma \approx 2.2517525890667211076474561638858515372118089180\ldots$

$H_\frac{3\pi}{2} \approx 2.2297778700699254440227548595030389742196711391976392264141982\ldots$

$\Psi^{(0)}(\frac{19}{2}) = \dfrac{\Gamma'(\frac{19}{2})}{\Gamma(\frac{19}{2})} = H_{\frac{17}{2}} - \gamma = \Psi^{(0)}(-\frac{17}{2}) = \dfrac{\Gamma'(-\frac{17}{2})}{\Gamma(-\frac{17}{2})} = H_{-\frac{19}{2}} - \gamma \\ \\ = \dfrac{3186538}{765765} - 2\ln 2 - \gamma \approx 2.1977378764029495331673303929550292101210294742\ldots$

$H_\frac{9}{2} = H_{-\frac{11}{2}} = \dfrac{1126}{315} \,-\, 2\ln 2 \approx 2.18830881348328398434013893168682146702360\ldots$

$\pi\ln2 \approx 2.177586090303602130500688898237613947338583700369286294325795\ldots$

$\Psi^{(0)}(9) = \dfrac{\Gamma'(9)}{\Gamma(9)} = \dfrac{\Gamma'(9)}{40320} = H_8 \! - \gamma = \Psi^{(0)}(-8) = \dfrac{\Gamma'(-8)}{\Gamma(-8)} = \\ \\ H_{-9} \,-\, \gamma = \dfrac{761}{280} \,-\, \gamma \approx 2.140641477955609996536345052774740426100697806917\ldots$

$\dfrac{\zeta(3)}{\xi(3)} = \dfrac{2\pi}{3} \approx 2.0943951023931954923084289221863352561314462662500705473\ldots$

$\Psi^{(0)}(\frac{17}{2}) = \dfrac{\Gamma'(\frac{17}{2})}{\Gamma(\frac{17}{2})} = H_{\frac{15}{2}} - \gamma = \Psi^{(0)}(-\frac{15}{2}) = \dfrac{\Gamma'(-\frac{15}{2})}{\Gamma(-\frac{15}{2})} = H_{-\frac{17}{2}} - \gamma \\ \\ = \dfrac{182144}{45045} - 2\ln 2 - \gamma \approx 2.08009081757942012140262451060208803365044123894\ldots$

$\cot\!\left(\dfrac{\pi}{7}\right) \approx 2.0765213965723365671635388614858403307057202066259685240834\ldots$

$\dfrac{\zeta'(-1)}{\zeta(-1)} - \dfrac{\xi'(-1)}{\xi(-1)} = \dfrac{1}{2} (1+\gamma+\ln(4\pi)) \\ \\ \approx 2.05411995593541182679220184217590713942022720878787283928037516267\ldots$

$2\,\dfrac{\Gamma(-\frac{1}{4})}{\Gamma(-\frac{3}{4})} = 6\,\dfrac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} = \dfrac{3\sqrt{2}}{\pi}\,(\Gamma(\frac{3}{4}))^2 = \dfrac{6\pi\sqrt{2}}{\left(\Gamma(\frac{1}{4})\right)^2} \\ \\ \approx 2.0279347202018541869863430540124172448618503647462178559305887420\ldots$

$\Psi^{(0)}(8) = \dfrac{\Gamma'(8)}{\Gamma(8)} = \dfrac{\Gamma'(8)}{5040} = H_7 \! - \gamma = \Psi^{(0)}(-7) = \dfrac{\Gamma'(-7)}{\Gamma(-7)} = \\ \\ H_{-8} \,-\, \gamma = \dfrac{363}{140} \,-\, \gamma \approx 2.015641477955609996536345052774740426100697806917\ldots$

$-\zeta''(0) = \dfrac{\ln^2(2\pi)}{2} + \dfrac{\pi^2}{24} - \dfrac{\gamma^2}{2} - \gamma_1 \\ \\ \approx 2.00635645590858485121010002672996043819899491016091988116986828085\ldots$

$\sec^2\!\left(\dfrac{\pi}{4}\right) = \csc^2\!\left(\dfrac{\pi}{4}\right) = 2\cot\!\left(\dfrac{\pi}{4}\right) = 2$

$\text{Li}_{-2}(\frac{1}{e}) = \dfrac{e(e+1)}{(e\!-\!1)^3} \\ \\ \approx 1.99229476712498739292601661300211738783140452306377006952350168484\ldots$

$\dfrac{\zeta'(-1)}{\zeta(-1)} = 12\ln{A}\,-\,1 \approx 1.98505372440541115056703592291336771316843656402\ldots$

$\zeta'(1) \,+\, \zeta(2) = \gamma^2 \,+\, \dfrac{\pi^2}{6} \approx 1.978111990655945110790791303001269415878367041\ldots$

$2\,\xi(6) = \dfrac{4\pi^3}{63} \approx 1.96865248763808382066516286140326318744287546450064175\ldots$

$H_\frac{7}{2} = H_{-\frac{9}{2}} = \dfrac{352}{105} \,-\, 2\ln 2 \approx 1.9660865912610617621179167094645992448013806\ldots$

$\Psi^{(0)}(\frac{15}{2}) = \dfrac{\Gamma'(\frac{15}{2})}{\Gamma(\frac{15}{2})} = H_{\frac{13}{2}} - \gamma = \Psi^{(0)}(-\frac{13}{2}) = \dfrac{\Gamma'(-\frac{13}{2})}{\Gamma(-\frac{13}{2})} = H_{-\frac{15}{2}} - \gamma \\ \\ = \dfrac{176138}{45045} - 2\ln 2 - \gamma \approx 1.94675748424608678806929117726875470031710790560\ldots$

$\text{Ei}(1) \approx 1.895117816355936755466520934331634269017060581732707591646228\ldots$

$\dfrac{3\ln2}{\ln3} \approx 1.892789260714372311298581343028282562898756920395641283611964\ldots$

$\dfrac{256\pi}{6302625A} \, 2^{-\frac{43}{12}} \, e^\frac{37}{3} \approx 1.8855734994646660069927789585135453448057679496677\ldots$

$\dfrac{3\sqrt{\pi}}{2\sqrt{2}} = \sqrt{\dfrac{9\pi}{8}} \approx 1.87997120597325037681182396360828393975524005545745373\ldots$

$\Psi^{(0)}(7) = \dfrac{\Gamma'(7)}{\Gamma(7)} = \dfrac{\Gamma'(7)}{720} = H_6 \! - \gamma = \Psi^{(0)}(-6) = \dfrac{\Gamma'(-6)}{\Gamma(-6)} = \\ \\ H_{-7} \,-\, \gamma = \dfrac{49}{20} \,-\, \gamma \approx 1.8727843350984671393934879099175975689578406640600\ldots$

$\sqrt{2+\!\sqrt{2}} \approx 1.847759065022573512256366378793576573644833251727284972230\ldots$

$\dfrac{\sinh(\pi)}{2\pi} \approx 1.8380389551874888603478487460141303332535781734138151387390\ldots$

$\dfrac{\zeta'(0)}{\zeta(0)} = \ln(2\pi) \approx 1.8378770664093454835606594728112352797227949472755668\ldots$

$\cot\!\left(\dfrac{1}{2}\right) = \dfrac{\sin 1}{1\!-\!\cos 1} \approx 1.83048772171245191926801943896881662375810794801\ldots$

$\Psi^{(0)}(\frac{13}{2}) = \dfrac{\Gamma'(\frac{13}{2})}{\Gamma(\frac{13}{2})} = H_{\frac{11}{2}} - \gamma = \Psi^{(0)}(-\frac{11}{2}) = \dfrac{\Gamma'(-\frac{11}{2})}{\Gamma(-\frac{11}{2})} = H_{-\frac{13}{2}} - \gamma \\ \\ = \dfrac{13016}{3465} - 2\ln 2 - \gamma \approx 1.792911330399932941915445023422600854163261751760\ldots$

$e^\gamma \approx 1.781072417990197985236504103107179549169645214303430205357665876\ldots$

$\theta_3\!\left(0,\frac{1}{e}\right) = \sqrt{\pi}\;\theta_3(0,e^{-\pi^2}) \\ \\ \approx 1.77263720482665215303125055115785848134338604537224605383159051087\ldots$

$\sqrt{\pi}\,(1+2e^{-\pi^2}) \\ \\ \approx 1.77263720482665212765975825886807357693113899487292155206947557284\ldots$

$\Gamma(\frac{1}{2}) = \sqrt{\pi} \approx 1.7724538509055160272981674833411451827975494561223871282\ldots$

$\dfrac{\theta_3(0,e^{-\frac{1}{2}})+1}{2} \approx 1.75331414402145277241533952693198018907372563575945498\ldots$

$\dfrac{\zeta'(-\frac{1}{2})}{\zeta(-\frac{1}{2})} \approx 1.7358261214241402631923577172095143758574132954684724804554\ldots$

$\dfrac{1}{\zeta(1)} = \dfrac{1}{\gamma} \approx 1.73245471460063347358302531586082968115577655226680502204\ldots$

$\sqrt{3} = \cot\!\left(\dfrac{\pi}{6}\right)\approx 1.73205080756887729352744634150587236694280525381038062\ldots$

$\xi(8) = \xi(-7) = \dfrac{4\pi^4}{225} \approx 1.73171717393782110642560591446586864443960152306\ldots$

$\dfrac{\xi(8)}{\zeta(8)} = \dfrac{168}{\pi^4} \approx 1.7246850187869683117776675888759406158270707114112281366\ldots$

$\Psi^{(0)}(6) = \dfrac{\Gamma'(6)}{\Gamma(6)} = \dfrac{\Gamma'(6)}{120} = H_5 \! - \gamma = \Psi^{(0)}(-5) = \dfrac{\Gamma'(-5)}{\Gamma(-5)} = \\ \\ H_{-6} \,-\, \gamma = \dfrac{137}{60} \,-\, \gamma \approx 1.706117668431800472726821243250930902291173997393\ldots$

$\dfrac{5}{2} \,-\, \dfrac{i}{2} \left(\Psi_1(i) - \Psi_1(-i)\right) \approx 1.70576645724068113441698638284309700409487089\ldots$

$\dfrac{\zeta'(1)}{\zeta(1)} = \dfrac{\zeta'(1)}{\gamma} = 1 + 2\ln(2\pi) - 12\ln A \\ \\ \approx 1.69070040841327981655428302270910284627715333052871224727033848564\ldots$

$\dfrac{\Gamma(-\frac{1}{6})}{\Gamma(-\frac{2}{3})} \approx 1.68542430561030885564995901780340869235155700777724816752744\ldots$

$H_\frac{5}{2} = H_{-\frac{7}{2}} = \dfrac{46}{15} \,-\, 2\ln 2 \approx 1.68037230554677604783220242375031353051566639\ldots$

$\text{erfi}(1) = -i \, \text{erf}(i) \approx 1.6504257587975428760253377295613624438956798748740\ldots$

$\dfrac{\zeta(4)}{\xi(4)} = \text{Li}_2(1) = \zeta(2) = \dfrac{\pi^2}{6} \approx 1.644934066848226436472415166646025189218949\ldots$

$\varphi = \dfrac{1+\sqrt{5}}{2} \approx 1.618033988749894848204586834365638117720309179805762862\ldots$

$\Psi^{(0)}(\frac{11}{2}) = \dfrac{\Gamma'(\frac{11}{2})}{\Gamma(\frac{11}{2})} = H_{\frac{9}{2}} - \gamma = \Psi^{(0)}(-\frac{9}{2}) = \dfrac{\Gamma'(-\frac{9}{2})}{\Gamma(-\frac{9}{2})} = H_{-\frac{11}{2}} - \gamma \\ \\ = \dfrac{1126}{315} - 2\ln 2 - \gamma \approx 1.6110931485817511237336268416044190359814435699427\ldots$

$\dfrac{2\sqrt{2}}{\sqrt{\pi}} \approx 1.595769121605730711759784239737527473903434524659738630663703\ldots$

$\dfrac{\ln3}{\ln2} \approx 1.5849625007211561814537389439478165087598144076924810604557526\ldots$

$(2\pi)^\frac{1}{4} \approx 1.583233487086159538579903034454558455661282237069305489373987\ldots$

$\text{Li}_{0}(\frac{1}{e}) = \dfrac{1}{1-e^{-1}} = \dfrac{e}{e-1} = \dfrac{1}{e-1} + 1 = \dfrac{1}{2}\!\left(\coth\left(\dfrac{1}{2}\right)\!+\!1\right) \\ \\ \approx 1.5819767068693264243850020051090115585468693010753961362667870596\ldots$

$2\,\xi(5) = \dfrac{15}{\pi^2}\,\zeta(5) \approx 1.57594121254077658394420640512441452870557565997071\ldots$

$\dfrac{\pi}{2} \approx 1.570796326794896619231321691639751442098584699687552910487472296\ldots$

$\tan 1 \approx 1.557407724654902230506974807458360173087250772381520038383946\ldots$

$\dfrac{1}{4}\coth\left(\dfrac{1}{2}\right) + 1 \approx 1.5409883534346632121925010025545057792734346505376980\ldots$

$\dfrac{\xi'(3)}{\xi(3)} + \dfrac{\zeta'(-2)}{\zeta(-2)} = \dfrac{1}{2} H_\frac{3}{2} - H_1 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = \dfrac{1}{3} + \dfrac{\ln\pi}{2} + \dfrac{\gamma}{2} \\ \\ \approx 1.540879698988772505416919114780152188715810474941078545553368491\ldots$

$\dfrac{\sqrt{2\pi}}{\sqrt{e}} \approx 1.520346901066280805611940146754975627036107418779046337528363\ldots$

$\dfrac{\zeta(2)}{\zeta(4)} = -\dfrac{3}{\pi^2} \, \dfrac{B_2}{B_4} = \dfrac{15}{\pi^2} \approx 1.5198177546350665716581919481459145835653816200\ldots$

$\ln\!\left(\dfrac{128}{9\pi}\right) = -\ln\pi \,+\, 7\ln 2 \,-\, 2\ln 3 \approx 1.51007580073399760898670702500912585\ldots$

$\Psi^{(0)}(5) = \dfrac{\Gamma'(5)}{\Gamma(5)} = \dfrac{\Gamma'(5)}{24} = H_4 \! - \gamma = \Psi^{(0)}(-4) = \dfrac{\Gamma'(-4)}{\Gamma(-4)} = \\ \\ H_{-5} \,-\, \gamma = \dfrac{25}{12} \,-\, \gamma \approx 1.5061176684318004727268212432509309022911739973934\ldots$

$\dfrac{1}{2}\ln(2\pi^2) = \ln(\pi\sqrt{2}) \approx 1.49130347612937282885204341208214699568504488009\ldots$

$\sqrt{6}\;\dfrac{\Gamma(-\frac{1}{3})}{\Gamma(-\frac{5}{6})} \approx 1.4897186945920968338888143005200445873066714109284026138\ldots$

$3\,-\,2H_\frac{2}{3} = 3\ln 3\,-\,\dfrac{\pi}{\sqrt{3}} \approx 1.482037501770111223591657453125421381658405425\ldots$

$\dfrac{(\Gamma(\frac{1}{4}))^2}{2\pi\sqrt{2}} = \dfrac{\pi}{\sqrt{2}\,(\Gamma(\frac{3}{4}))^2} \approx 1.4793375595943194461554106788638597832374490980\ldots$

$\lambda_8 \approx 1.46575567714706063265551454197774878791984786187454446584585775\ldots$

$\dfrac{\sqrt{\pi}}{2} \, \text{erfi}(1) \approx 1.46265174590718160880404858685698815512087009621673918566\ldots$

$\dfrac{1}{2} \,+\, \dfrac{1}{e} \,+\, \dfrac{1}{e-1} \approx 1.44985614804076874598052577527047242599268043210716397\ldots$

$e^\frac{1}{e} \approx 1.44466786100976613365833910859643022305859545324225316582052266\ldots$

$\dfrac{2^\frac{1}{4}\,\sqrt{6\pi}}{\Gamma(\frac{1}{4})} \approx 1.4240557293174499169777816343531653993726123689987929253244\ldots$

$\dfrac{\zeta'(2)}{\zeta(2)} + \dfrac{\zeta'(-1)}{\zeta(-1)} = -H_1 + \ln(2\pi) + \gamma = -1 + \ln(2\pi) + \gamma \\ \\ \approx 1.41509273131087834416717156289363771076495428321549042444007031585\ldots$

$\csc\!\left(\dfrac{\pi}{4}\right) = \sqrt{2} \approx 1.4142135623730950488016887242096980785696718753769480\ldots$

$\sqrt{\frac{\pi e}{2}} \, \text{erf}\left(\frac{1}{\sqrt{2}}\right) \approx 1.41068613464244799769082471141911504132347862562519219\ldots$

$\Psi^{(0)}(\frac{9}{2}) = \dfrac{\Gamma'(\frac{9}{2})}{\Gamma(\frac{9}{2})} = H_{\frac{7}{2}} - \gamma = \Psi^{(0)}(-\frac{7}{2}) = \dfrac{\Gamma'(-\frac{7}{2})}{\Gamma(-\frac{7}{2})} = H_{-\frac{9}{2}} - \gamma \\ \\ = \dfrac{352}{105} - 2\ln 2 - \gamma \approx 1.38887092635952890151140461938219681375922134772051\ldots$

$\dfrac{1}{2}\left(\theta_3\!\left(0,\dfrac{1}{e}\right)+1\right) \approx 1.386318602413326076515625275578929240671693022686\ldots$

$2\,-\,H_\frac{1}{2} = 2\ln 2 \approx 1.386294361119890618834464242916353136151000268720510\ldots$

$\dfrac{\sqrt{\pi}+1}{2} \approx 1.3862269254527580136490837416705725913987747280611935641069\ldots$

$\sqrt{1+\dfrac{2}{\sqrt{5}}} \approx 1.37638192047117353820720958191088767952589933600815866336\ldots$

$\Gamma(\frac{2}{3}) = \dfrac{2\pi}{\sqrt{3}\,\Gamma(\frac{1}{3})} \approx 1.354117939426400416945288028154513785519327266056793\ldots$

$\dfrac{(2\pi)^\frac{1}{4}}{A\,e^{-\frac{1}{12}}} \approx 1.34184847575580468974435361708077867699012703621467521803484\ldots$

$\zeta(\frac{5}{2}) \approx 1.3414872572509171797567696933486121366230376295059865112537967\ldots$

$2^{\frac{5}{12}} \approx 1.33483985417003436483083188118445277491239021262519829693897081\ldots$

$M_3 \approx 1.33258227573322088176582877607102774883845948904242266178713089\ldots$

$\Gamma(\frac{5}{2}) = \dfrac{3\sqrt{\pi}}{4} \approx 1.329340388179137020473625612505858887098162092091790346\ldots$

$\dfrac{5}{3} - \dfrac{2}{3}\,H_\frac{2}{5} = \dfrac{\pi}{3}\,\sqrt{1-\dfrac{2}{\sqrt{5}}}\,+\,\dfrac{5\ln 5}{6}\,-\,\dfrac{\sqrt{5}}{6}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right) \\ \\ \approx 1.32277925312238885674944226131008401652280117371392437228545762688\ldots$

$\text{Ei}(1) \,-\, \gamma \approx 1.317902151454403894860008844249231837974901245792783992840\ldots$

$2\,\xi(4) = \dfrac{12}{\pi^2}\,\zeta(4) = \dfrac{\zeta(5)}{\xi(5)} = \dfrac{2\pi^2}{15} \\ \\ \approx 1.31594725347858114917793213331682015137515992096543875018844658349\ldots$

$H_\frac{\pi}{2} \approx 1.31450881369651659005000275761275678370798149959118894812400897\ldots$

$\dfrac{\varpi}{2} = \dfrac{\left(\Gamma(\frac{1}{4})\right)^2}{4\sqrt{2\pi}} \approx 1.31102877714605990523241979494555970684137747571581158\ldots$

$\lambda \approx 1.303577269034296391257099112152551890730702504659404875754861390\ldots$

$\displaystyle \int_0^1 \dfrac{dx}{x^x} \approx 1.29128599706266354040728259059560054149861936827452231731000\ldots$

$\dfrac{1}{2}\,+\,\dfrac{\pi}{4} \approx 1.28539816339744830961566084581987572104929234984377645524373\ldots$

$A = e^{\frac{1}{12}\,-\,\zeta'(-1)} \approx 1.2824271291006226368753425688697917277676889273250011\ldots$

$H_\frac{\pi}{2} + H_{-\frac{\pi}{2}} \approx 1.28191194432037469488687537026274344240529332646466119132\ldots$

$\xi(7) = \xi(-6) = \dfrac{315}{8\pi^3}\,\zeta(7) \approx 1.2805069504568869598383653329720308646010482\ldots$

$H_\frac{3}{2} = H_{-\frac{5}{2}} = \dfrac{8}{3} \,-\, 2\ln 2 \approx 1.28037230554677604783220242375031353051566639\ldots$

$\dfrac{3}{2} \,-\, \dfrac{1}{2}\,H_\frac{1}{3} = \dfrac{\pi}{4\sqrt{3}} \,+\, \dfrac{3\ln 3}{4} \approx 1.2774090575596367311949534921024332115566344\ldots$

$\dfrac{\xi(7)}{\zeta(7)} = \dfrac{315}{8\pi^3} \approx 1.269904168307229886636618324612195238351868860806949271\ldots$

$\ln\!\left(\dfrac{9\pi}{8}\right) = \ln\pi \,-\, 3\ln 2 \,+\, 2\ln 3 \approx 1.26251292150578362868222146082358041671\ldots$

$\dfrac{2\ln 2}{\ln 3} \approx 1.261859507142914874199054228685521708599171280263760855741309\ldots$

$\dfrac{\zeta'(0)}{\zeta(0)} - \dfrac{\zeta'(1)}{\zeta(1)} = \ln(2\pi) - \gamma \\ \\ \approx 1.26066140150781262295414738272883284868063561133564322682853584608\ldots$

$\Psi^{(0)}(4) = \dfrac{\Gamma'(4)}{\Gamma(4)} = \dfrac{\Gamma'(4)}{6} = H_3 \! - \gamma = \Psi^{(0)}(-3) = \dfrac{\Gamma'(-3)}{\Gamma(-3)} = \\ \\ H_{-4} \,-\, \gamma = \dfrac{11}{6} \,-\, \gamma \approx 1.2561176684318004727268212432509309022911739973934\ldots$

$\dfrac{\sqrt{2\pi}}{2} = \dfrac{\sqrt{\pi}}{\sqrt{2}} = \sqrt{\dfrac{\pi}{2}} \approx 1.253314137315500251207882642405522626503493370304\ldots$

$e(1\,-\,\ln(e-1)) \approx 1.2468083128715153703801373560103055352504978363119469\ldots$

$A^2 \, (2e)^{-\frac{1}{6}} \approx 1.24025593474508628719781649988411902623556725730644159018\ldots$

$\dfrac{\pi^2}{8} \approx 1.23370055013616982735431137498451889191421242590509882830166867\ldots$

$\ln\!\left(\tan\left(\dfrac{\pi}{4}+\dfrac{1}{2}\right)\!\right) = -\ln\!\left(\tan\left(\dfrac{\pi}{4}-\dfrac{1}{2}\right)\!\right) = -\ln(\sec1 - \tan1) \\ \\ \approx 1.22619117088351707081306096747190675272424835022074027913861684\ldots$

$\Gamma(\frac{3}{4}) = \dfrac{\pi\sqrt{2}}{\Gamma(\frac{1}{4})}\approx 1.225416702465177645129098303362890526851239248108070611\ldots$

$\dfrac{4}{3} \,-\, \dfrac{1}{3}\,H_\frac{1}{4} = \dfrac{\pi}{6} \,+\, \ln 2 \approx 1.21674595615824418249433935200476038210836170092\ldots$

$\zeta(3) = 24 \, \zeta(2) \, \ln A_2 = 4\pi^2 \, \ln A_2 \\ \\ \approx 1.20205690315959428539973816151144999076498629234049888179227155534\ldots$

$A \, e^{-\frac{1}{12}} \approx 1.17988991729814590705100799045229956719253920355131915641598\ldots$

$\dfrac{5}{4} - \dfrac{1}{4}\,H_\frac{1}{5} = \dfrac{\pi}{8}\,\sqrt{1+\dfrac{2}{\sqrt{5}}}\,+\,\dfrac{5\ln 5}{16}\,+\,\dfrac{\sqrt{5}}{16}\,\ln\!\left(\dfrac{3+\sqrt{5}}{2}\right) \\ \\ \approx 1.17795605792266385873517396809188741844585723456667980284252285732\ldots$

$\dfrac{\pi}{e} \approx 1.155727349790921717910093183312696299120851023164415820499706535\ldots$

$\dfrac{\zeta(1)}{\xi(1)} = 2\gamma \approx 1.15443132980306572121302418016480486208431867187984719761\ldots$

$2\,\xi(3) = \dfrac{3}{\pi}\,\zeta(3) \approx 1.147879788093511026750459420722598411077951185651400\ldots$

$\dfrac{\zeta'(1)}{\zeta(1)} - \dfrac{\zeta'(2)}{\zeta(2)} = - \ln(2\pi) + 12 \ln A \\ \\ \approx 1.14717665799606566700637645010213243344564161674685457836396459532\ldots$

$\zeta(2) - \dfrac{1}{2} = \dfrac{\pi^2}{6} \,-\, \dfrac{1}{2} \approx 1.1449340668482264364724151666460251892189499012067\ldots$

$e\!\left(1-\dfrac{1}{e\!-\!1}\!\right) \approx 1.13630512158971881097528546624365093921037779262456343\ldots$

$\dfrac{1}{2}(\theta_3(0,e^{-2})-1) \approx 1.135670761094507612611191289395467812488432493858813\ldots$

$\dfrac{2}{\sqrt{\pi}} \approx 1.1283791670955125738961589031215451716881012586579977136881714\ldots$

$\zeta(\frac{7}{2}) \approx 1.1267338673170566464278124918549842722219969574036029638423960\ldots$

$\lambda_7 \approx 1.12446011757095949058282010801697564045977094323138314124840761\ldots$

$2^{\frac{1}{6}} \approx 1.1224620483093729814335330496791795162324111106139867534404095\ldots$

$\Psi^{(0)}(\frac{7}{2}) = \dfrac{\Gamma'(\frac{7}{2})}{\Gamma(\frac{7}{2})} = H_{\frac{5}{2}} - \gamma = \Psi^{(0)}(-\frac{5}{2}) = \dfrac{\Gamma'(-\frac{5}{2})}{\Gamma(-\frac{5}{2})} = H_{-\frac{7}{2}} - \gamma \\ \\ = \dfrac{46}{15} - 2\ln 2 - \gamma \approx 1.103156640645243187225690333667911099473507062006232\ldots$

$\sqrt{\pi} \, \text{erfi}\left(\dfrac{1}{2}\right) \approx 1.089974208367244447324840262814033342081596207257870379\ldots$

$\theta_3(0,e^{-\pi}) \approx 1.08643481121330801457531612151022345707020570724521888592\ldots$

$\zeta(4) = \dfrac{\pi^4}{90} \approx 1.08232323371113819151600369654116790277475095191872690768\ldots$

$\dfrac{1}{e-1} \,+\, \dfrac{1}{2} = \dfrac{1}{2} \coth\left(\dfrac{1}{2}\right) \approx 1.0819767068693264243850020051090115585468693\ldots$

$\dfrac{\zeta'(-2)}{\zeta(-2)} \approx 1.0799154134691555843536649576931127757879638636378936953118\ldots$

$\dfrac{\Gamma(\frac{1}{3})}{\sqrt{2\pi}} = \dfrac{\sqrt{2\pi}}{\sqrt{3}\,\Gamma(\frac{2}{3})} \approx 1.06874184809158144465945314257609713381264422464077\ldots$

$\dfrac{\zeta(4)}{\zeta(6)} = -\dfrac{15}{2\pi^2} \, \dfrac{B_4}{B_6} = \dfrac{21}{2\pi^2} \approx 1.06387242824454660016073436370214020849576713\ldots$

$\zeta(\frac{9}{2}) \approx 1.0547075107614542640229672889602801172724938329562517306846845\ldots$

$\dfrac{1}{2}(\theta_3(0,e^{-3})-1) \approx 1.049793212582096800007065345335742466325680137917424\ldots$

$2\,\xi(2) = \dfrac{2}{\pi}\,\zeta(2) = \dfrac{\pi}{3} \approx 1.04719755119659774615421446109316762806572313312\ldots$

$\dfrac{\gamma}{3} \,+\, \dfrac{\pi}{6\sqrt{3}} \,+\, \dfrac{\ln 3}{2} \approx 1.044011260006935440998806358095756284718476098906806\ldots$

$\zeta(5) = -120 \, \zeta(4) \, \ln A_4 = -\dfrac{4\pi^4}{3} \, \ln A_4 \\ \\ \approx 1.03692775514336992633136548645703416805708091950191281197419267790\ldots$

$M_2 \approx 1.03465388189743791161979429846463825467030798434438525450307028\ldots$

$\dfrac{\zeta(6)}{\xi(6)} = \dfrac{\pi^3}{30} \approx 1.03354255600999400584921050223671317340750961886283692313\ldots$

$A_2 = e^{-\zeta'(-2)} = e^{\frac{\zeta(3)}{4\pi^2}} \approx 1.03091675219739211419331309646694229063319430640\ldots$

$A_{10} = e^{-\zeta'(-10)} = e^{\frac{14175\zeta(11)}{8\pi^{10}}} \approx 1.019110233329383853722164704986297513513481\ldots$

$A_9 = e^{\frac{7129}{332640} - \zeta'(-9)} \approx 1.0184699299209929121706590493766721723086101905640\ldots$

$\zeta(6) = \dfrac{\pi^6}{945} \approx 1.0173430619844491397145179297909205279018174900328535618\ldots$

$\dfrac{\zeta(6)}{\zeta(8)} = -\dfrac{14}{\pi^2} \, \dfrac{B_6}{B_8} = \dfrac{10}{\pi^2} \approx 1.0132118364233777144387946320972763890435877467\ldots$

$A_5 = e^{\frac{137}{15120} - \zeta'(-5)} \approx 1.00968038728586616112008919046263069260327634721152\ldots$

$\zeta(7) = 168 \, \zeta(6) \, \ln A_6 = \dfrac{8\pi^6}{45} \, \ln A_6 \\ \\ \approx 1.00834927738192282683979754984979675959986356056523870641728313657\ldots$

$A_6 = e^{-\zeta'(-6)} = e^{\frac{15\zeta(7)}{4\pi^6}} \approx 1.0059171969986734684440139835542556563906156550\ldots$

$\zeta(8) = \dfrac{\pi^8}{9450} \approx 1.004077356197944339378685238508652465258960790649850020\ldots$

$\dfrac{\zeta(8)}{\zeta(10)} = -\dfrac{45}{2\pi^2} \, \dfrac{B_8}{B_{10}} = \dfrac{99}{10\pi^2} \approx 1.00307971805914393729440668577630362515315\ldots$

$\zeta(9) = -120 \, \zeta(8) \, \ln A_8 = -\dfrac{4\pi^8}{315} \, \ln A_8 \\ \\ \approx 1.00200839282608221441785276923241206048560585139488875654859661590\ldots$

$\zeta(10) = \dfrac{\pi^{10}}{93555} \approx 1.0009945751278180853371459589003190170060195315644775\ldots$

$\theta_3(0,e^{-\pi^2}) \approx 1.00010344637240763892662260055284706012766757564311305216\ldots$

$\dfrac{\eta(0)}{\xi(0)} = \dfrac{\xi(1)}{\xi(0)} = \cos(0) = \sec(0) = \sin\!\left(\dfrac{\pi}{2}\right)\! = \csc\!\left(\dfrac{\pi}{2}\right)\! = \cot\!\left(\dfrac{\pi}{4}\right)\! = \nolinebreak H_1 = \nolinebreak 1$

$\eta(10) = \dfrac{73\,\pi^{10}}{6842880} \approx 0.99903950759827156563922184569934183142592964966689\ldots$

$\eta(9) = \dfrac{255\,\zeta(9)}{256} \approx 0.99809429754160533076778303185259795087433395353787\ldots$

$\dfrac{523833}{524875} \approx 0.9980147654203381757561324124791617051678971183615146463443\ldots$

$\dfrac{\zeta(9)}{\zeta(8)} = \dfrac{4}{B_8} \, \ln A_8 = -120 \ln A_8 \\ \\ \approx 0.99793943827226968314293118126410570708089475025959988482526393036\ldots$

$\dfrac{\zeta(10)}{\zeta(8)} = -\dfrac{2\pi^2}{45} \, \dfrac{B_{10}}{B_8} = \dfrac{10\pi^2}{99} \approx 0.99692973748377359786206979796728799346603\ldots$

$\eta(8) = \dfrac{127\,\pi^8}{1209600} \approx 0.996233001852647899227289260082803617874125159472898\ldots$

$\dfrac{\zeta(18)}{\zeta(9)^2} = \dfrac{43867\;\pi^{18}}{38979295480125\;\zeta(9)^2} \approx 0.99599908495575988109926339437836556066\ldots$

$\eta(7) = \dfrac{63\,\zeta(7)}{64} \approx 0.9925938199228302826704257131333936852311156924314068\ldots$

$A_4 = e^{-\zeta'(-4)} = e^{-\frac{3\zeta(5)}{4\pi^4}} \approx 0.992047974525040260013436977625443356736904851\ldots$

$\dfrac{7234}{7293} \approx 0.991910050733580145344851227204168380638968874262991910050733\ldots$

$A_8 = e^{-\zeta'(-8)} = e^{\frac{315\zeta(9)}{4\pi^8}} \approx 0.991718321632822196999547482765793339867859760\ldots$

$\dfrac{\zeta(7)}{\zeta(6)} = \dfrac{4}{B_6} \, \ln A_6 = 168 \ln A_6 \\ \\ \approx 0.99115953611067749170581940460986029740665781618640890536021789066\ldots$

$A_7 = e^{-\frac{121}{11120} - \zeta'(-7)} \approx 0.9899756533334170941753964830588692002082471514307\ldots$

$\dfrac{\zeta(8)}{\zeta(6)} = -\dfrac{\pi^2}{14} \, \dfrac{B_8}{B_6} = \dfrac{\pi^2}{10} \approx 0.9869604401089358618834490999876151135313699407\ldots$

$\eta(6) = \dfrac{31\,\pi^6}{30240} \approx 0.9855510912974351040984392444849542614048856934693268\ldots$

$\xi(6) = \xi(-5) = \dfrac{2\pi^3}{63} \approx 0.98432624381904191033258143070163159372143773225\ldots$

$\dfrac{\zeta(14)}{\zeta(7)^2} = \dfrac{2\pi^{14}}{18243225\;\zeta(7)^2} \approx 0.98356851051172800793614495902996938666466436\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{-\frac{1}{2}}}{2} = \dfrac{\gamma}{2} \,+\, \ln 2 \approx 0.981755013010711739720488166499377783596579802\ldots$

$A_3 = e^{-\frac{11}{720} - \zeta'(-3)} \approx 0.97955552694284460582421883726349182644553675249552\ldots$

$\zeta'(1) = \gamma \, (1 + 2\ln(2\pi) - 12\ln A) \\ \\ \approx 0.9758987603915644713556011204053297460809201353635083940699601399\ldots$

$\eta(5) = \dfrac{15\,\zeta(5)}{16} \approx 0.972119770446909305935655143553469532553513362033043\ldots$

$\dfrac{\xi(6)}{\zeta(6)} = \dfrac{30}{\pi^3} \approx 0.96754603299598467553266158065691065779190008442434230203\ldots$

$\dfrac{691}{715} \approx 0.9664335664335664335664335664335664335664335664335664335664335\ldots$

$\dfrac{\zeta(5)}{\zeta(4)} = \dfrac{4}{B_4} \, \ln A_4 = -120 \ln A_4 \\ \\ \approx 0.95805737403223491368360049585471646862852320347586517012456876568\ldots$

$\eta(4) = \dfrac{7\,\pi^4}{720} \approx 0.947032829497245917576503234473521914927907082928886044\ldots$

$2^{-\frac{1}{12}} \approx 0.9438743126816934966419131566675343760075683033387428137421251\ldots$

$\dfrac{\zeta(6)}{\zeta(4)} = -\dfrac{2\pi^2}{15} \, \dfrac{B_6}{B_4} = \dfrac{2\pi^2}{21} \approx 0.93996232391327224941280866665487153669654280\ldots$

$-\zeta'(2) = \zeta(2) \, (12\ln A - \ln(2\pi) - \zeta(1)) = \dfrac{\pi^2}{6} \, (12\ln A - \ln(2\pi) - \gamma) \\ \\ \approx 0.93754825431584375370257409456786497789786028861482992588543348036\ldots$

$\ln\!\left(\dfrac{8}{\pi}\right) = -\ln\pi \,+\, 3\ln 2 \approx 0.93471165583043575410826901302147099257920559\ldots$

$\dfrac{\zeta(10)}{\zeta(5)^2} = \dfrac{\pi^{10}}{93555\;\zeta(5)^2} \approx 0.93096793995852028403372887222171855467361752106\ldots$

$e^{\gamma_1} \approx 0.92977203598071089745654153195024214142992586769967109903152039\ldots$

$2\tanh\!\left(\dfrac{1}{2}\right) \approx 0.9242343145200195170046369672873450974605785606602260771\ldots$

$\Psi^{(0)}\!(3) = \dfrac{\Gamma'(3)}{\Gamma(3)} =\!H_2-\gamma = \Psi^{(0)}\!(-2) = \dfrac{\Gamma'(-2)}{\Gamma(-2)} = H_{-3}-\gamma = \nolinebreak \dfrac{3}{2}-\gamma \\ \\ \approx 0.92278433509846713939348790991759756895784066406007640119423276511\ldots$

$\dfrac{(2\pi)^2}{\cosh(\pi\sqrt{2}) - \cos(\pi\sqrt{2})} \approx 0.92278136647696324668391153344904637588225782\ldots$

$-2\left(1\,+\,\zeta(\frac{1}{2})\right) \approx 0.9207090176191736257789983050305960249344586620251629\ldots$

$\text{Li}_{-1}(\frac{1}{e}) = \dfrac{e}{(e\!-\!1)^2} \\ \\ \approx 0.92067359420779231894541352271649960288165562665055115235396040972\ldots$

$e^{-\frac{1}{12}} \approx 0.9200444146293232478931553240537172316731875349599742017014157\ldots$

$\Gamma\!\left(\dfrac{7}{4}\right) = \dfrac{3\pi}{2\sqrt{2}\,\Gamma(\frac{1}{4})} \approx 0.919062526848883233846823727522167895138429436081\ldots$

$\dfrac{\sinh(\pi)}{4\pi} \approx 0.9190194775937444301739243730070651666267890867069075693695\ldots$

$-\zeta'(0) = \dfrac{\ln(2\pi)}{2} \approx 0.91893853320467274178032973640561763986139747363778\ldots$

$C = \beta(2) \approx 0.91596559417721901505460351493238411077414937428167213426\ldots$

$\dfrac{1}{2}\cot\!\left(\dfrac{1}{2}\right) = \dfrac{\sin 1}{2(1\!-\!\cos 1)} \approx 0.9152438608562259596340097194844083118790539\ldots$

$\dfrac{\zeta'(3)}{\zeta(3)} + \dfrac{\zeta'(-2)}{\zeta(-2)} = -H_2 + \ln(2\pi) + \gamma = -\dfrac{3}{2} + \ln(2\pi) + \gamma \\ \\ \approx 0.91509273131087834416717156289363771076495428321549042444007031585\ldots$

$\Gamma\!\left(\dfrac{5}{4}\right) = \dfrac{1}{4}\,\Gamma\!\left(\dfrac{1}{4}\right) \approx 0.9064024770554770779826712889669180007487919207200\ldots$

$2\ln\!\left(\dfrac{\pi}{2}\right) \approx 0.903165410578909729452390459789764287143589357110112634785\ldots$

$\eta(3) = \dfrac{3\,\zeta(3)}{4} \approx 0.90154267736969571404980362113358749307373971925537416\ldots$

$\dfrac{\sqrt{3}+1}{12\sqrt{2}}\,\Gamma(\frac{1}{6}) \approx 0.896108063900939431713641110521645004781542249281857177\ldots$

$\Gamma(\frac{4}{3}) = \dfrac{1}{3}\,\Gamma(\frac{1}{3}) \approx 0.8929795115692492112185643136582258813762297926524337\ldots$

$\dfrac{2\sqrt{e}}{e+1} \approx 0.886818883970073908658897797783408562534089088712613924836258\ldots$

$\Gamma(\frac{3}{2}) = \dfrac{\sqrt{\pi}}{2} \approx 0.8862269254527580136490837416705725913987747280611935641\ldots$

$\text{arsinh}(1) = \ln(1+\sqrt{2}) \approx 0.881373587019543025232609324979792309028160328\ldots$

$\Psi^{(0)}(\frac{3}{4}) - \Psi^{(0)}(\frac{1}{2})= \dfrac{\Gamma'(\frac{3}{4})}{\Gamma(\frac{3}{4})} - \dfrac{\Gamma'(\frac{1}{2})}{\Gamma(\frac{1}{2})} = \dfrac{\pi}{2}\,-\,\ln 2 \\ \\ \approx 0.87764914623495130981408957018157487402308456532729765636679228666\ldots$

$\dfrac{\sqrt{5+\sqrt{5}}}{10\sqrt{2}}\,\Gamma(\frac{1}{5}) \approx 0.873230365517818589715164160575861054313944165712022\ldots$

$\dfrac{\xi(1)}{\zeta(1)} = \dfrac{1}{2\gamma} \approx 0.8662273573003167367915126579304148405778882761334025110\ldots$

$H_\frac{4}{5} = H_{-\frac{4}{5}} + \dfrac{5}{4} - \pi\cot(\frac{4\pi}{5}) = \dfrac{5}{4} + \dfrac{\pi}{2} \sqrt{1+\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} - \dfrac{\sqrt{5}}{4} \ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx 0.8622070981953944012152144569255670114079988462447007696959564406\ldots$

$\dfrac{\zeta(8)}{\zeta(4)^2} = \dfrac{6}{7} \approx 0.857142857142857142857142857142857142857142857142857142\ldots$

$-\dfrac{1}{3} \, H_{-\frac{2}{3}} = \dfrac{\pi}{6\sqrt{3}} \,+\, \dfrac{\ln 3}{2} \approx 0.851606038373091154129968994734955474371089653\ldots$

$\dfrac{\pi \sinh(\pi)}{\cosh(\pi\sqrt{2}) - \cos(\pi\sqrt{2})} \approx 0.84805404935290039212965018340500770584798748\ldots$

$\text{erf}\,(1) \approx 0.84270079294971486934122063508260925929606699796630290845993\ldots$

$\sin1 \approx 0.8414709848078965066525023216302989996225630607983710656727517\ldots$

$\dfrac{\sqrt{2+\sqrt{2}}}{8}\,\Gamma(\frac{1}{4}) \approx 0.837406696769086483083602722180832261379061661299010\ldots$

$\dfrac{\pi}{3\sqrt{3}} \,+\, \dfrac{\ln 2}{3} \approx 0.8356488482647210533371034597001107667865221274843319432\ldots$

$\dfrac{\varpi}{\pi} = \dfrac{1}{M(1,\sqrt{2})} = \left(\dfrac{\Gamma(\frac{1}{4})}{(2\pi)^\frac{3}{4}}\right)^2 \! \approx 0.8346268416740731862814297327990468089939\ldots$

$\lambda_6 \approx 0.82756601228237929742500282202049998136833796471926900594048579\ldots$

$M_0 = \dfrac{1}{4} \,+\, \gamma \approx 0.827215664901532860606512090082402431042159335939923598\ldots$

$H_\frac{3}{4} = H_{-\frac{3}{4}} + \dfrac{4}{3} - \pi\cot(\frac{3\pi}{4}) = \dfrac{4}{3} + \dfrac{\pi}{2} - 3\ln 2 \\ \\ \approx 0.82468811844839402431295866059855507120541762994012048145876560100\ldots$

$2\pi \, \text{sech}\left(\dfrac{\pi\sqrt{3}}{2}\right) \approx 0.82365884516434231810338374974455804503165201268072\ldots$

$\eta(2) = -\text{Li}_2(-1) = \dfrac{\pi^2}{12} \approx 0.822467033424113218236207583323012594609474950\ldots$

$\dfrac{\gamma}{3} \,-\, \dfrac{\pi}{6\sqrt{3}} \,+\, \dfrac{\ln 3}{2} \,-\, \dfrac{\ln 2}{3} \approx 0.812962199820287004526395650943030762026642720\ldots$

$\dfrac{1}{3\pi} \, \cosh\left(\dfrac{\pi\sqrt{3}}{2}\right) \approx 0.809396597366290109578680478726382119372787648261\ldots$

$\dfrac{\xi'(4)}{\xi(4)} + \dfrac{\zeta'(-3)}{\zeta(-3)} = -\dfrac{1}{2} H_2 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = -\dfrac{3}{4} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} \\ \\ \approx 0.80411995593541182679220184217590713942022720878787283928037516267\ldots$

$\dfrac{\sqrt{2}}{\sqrt{\pi}} = \sqrt{\dfrac{2}{\pi}} \approx 0.7978845608028653558798921198687637369517172623298693153\ldots$

$2^{-\frac{1}{3}} \approx 0.7937005259840997373758528196361541301957466639499265049041428\ldots$

$\dfrac{1+\zeta(1)}{2} = \dfrac{1+\gamma}{2} \approx 0.7886078324507664303032560450412012155210796679699\ldots$

$\xi(5) = \xi(-4) = \dfrac{15}{2\pi^2}\,\zeta(5) \approx 0.7879706062703882919721032025622072643527878\ldots$

$\dfrac{\zeta(7)}{\xi(7)} = \dfrac{8\pi^3}{315} \approx 0.787460995055233528266065144561305274977150185800256703\ldots$

$\dfrac{\pi}{4} \approx 0.785398163397448309615660845819875721049292349843776455243736148\ldots$

$\displaystyle \int_0^1 x^x dx \approx 0.783430510712134407059264386526975469407681990146930958255\ldots$

$\dfrac{A\,e^\frac{5}{12}}{\sqrt{2\pi}} \approx 0.776066249400476536921955776494057515374796747531129272035701\ldots$

$\dfrac{1}{2\sqrt{3}}\,\Gamma(\frac{1}{3}) \approx 0.77334294207798985019610161129521734092480684722421567266\ldots$

$M_2 \,-\, M_1 \approx 0.773156669049795127864367459855942395618741336083186048311\ldots$

$\dfrac{\xi(5)}{\zeta(5)} = \dfrac{15}{2\pi^2} \approx 0.7599088773175332858290959740729572917826908100418491163\ldots$

$H_\frac{2}{3} = H_{-\frac{2}{3}} + \dfrac{3}{2} - \pi\cot(\frac{2\pi}{3}) = \dfrac{3}{2} + \dfrac{\pi}{2\sqrt{3}} - \dfrac{3\ln 3}{2} \\ \\ \approx 0.7589812491149443882041712734372893091707972873122461101829009672\ldots$

$M_3 \,-\, \gamma \approx 0.7553666108316880211593166859886253177963001531024990629813\ldots$

$\ln\!\left(\dfrac{8}{3}\sqrt{\dfrac{2}{\pi}}\right) = -\dfrac{\ln\pi}{2} \,+\, \dfrac{7\ln 2}{2} \,-\, \ln 3 \approx 0.75503790036699880449335351250456292\ldots$

$\dfrac{\theta_3(0,e^{-\frac{1}{2}})-1}{2} \approx 0.75331414402145277241533952693198018907372563575945498\ldots$

$\dfrac{\sqrt{\pi}}{2} \, \text{erf}\,(1) \approx 0.74682413281242702539946743613185300535449968681260632902\ldots$

$-\dfrac{\pi}{2\sqrt{3}} \,+\, \dfrac{3\ln 3}{2} \approx 0.7410187508850556117958287265627106908292027126877538\ldots$

$\ln(\sinh(1))\,+\,\gamma \approx 0.7386550264727284942166318185247518030049613805578225\ldots$

$\dfrac{e}{1+e} = 1\,-\,\dfrac{1}{1+e} \approx 0.7310585786300048792511592418218362743651446401650\ldots$

$\dfrac{\zeta(3)}{\zeta(2)} = \dfrac{4}{B_2} \, \ln A_2 = 24 \ln A_2 \\ \\ \approx 0.73076296940143849872603673130771463952801160507937447007132535661\ldots$

$e^{-\gamma^2} \approx 0.7166426750181771686285534605583497355485794097592999041423739\ldots$

$\dfrac{\sqrt{2}}{2} = \dfrac{1}{\sqrt{2}} \approx 0.70710678118654752440084436210484903928483593768847403658\ldots$

$\dfrac{\zeta(6)}{\zeta(3)^2} = \dfrac{\pi^6}{945\;\zeta(3)^2} \approx 0.7040724873207844782962981999786244580925837811199\ldots$

$H_\frac{3}{5} = H_{-\frac{3}{5}} + \dfrac{5}{3} - \pi\cot(\frac{3\pi}{5}) = \dfrac{5}{3} + \dfrac{\pi}{2} \sqrt{1-\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} + \dfrac{\sqrt{5}}{4} \ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx 0.7032631176750091125125148079428371561982917479169830572057486382\ldots$

$\Psi^{(0)}(\frac{5}{2}) = \dfrac{\Gamma'(\frac{5}{2})}{\Gamma(\frac{5}{2})} = H_{\frac{3}{2}} - \gamma = \Psi^{(0)}(-\frac{3}{2}) = \dfrac{\Gamma'(-\frac{3}{2})}{\Gamma(-\frac{3}{2})} = H_{-\frac{5}{2}} - \gamma \\ \\ = \dfrac{8}{3} - 2\ln 2 - \gamma \approx 0.7031566406452431872256903336679110994735070620062325\ldots$

$\sqrt{2} \; \xi\left(\frac{1}{2}\right) = -\dfrac{\Gamma\left(\frac{1}{4}\right) \, \zeta\left(\frac{1}{2}\right)}{4\sqrt{2}\,\pi^{\frac{1}{4}}} \approx 0.7030349466513809051531057880214525054267304\ldots$

$\dfrac{\Gamma(\frac{1}{4})}{2^\frac{1}{4}\,\sqrt{6\pi}} \approx 0.7022197091115950362096834668744057545145562895011317104618\ldots$

$\dfrac{\text{Ei}(1)}{e} \approx 0.697174883235066068765478681919551595317175430954369517320054\ldots$

$\eta(2) = \ln 2 \approx 0.6931471805599453094172321214581765680755001343602552541\ldots$

$\dfrac{\pi}{4\sqrt{2}\,(\Gamma(\frac{5}{4}))^2} = \dfrac{\sqrt{2}}{\pi\,(\Gamma(\frac{3}{4}))^2} = \dfrac{2\pi\sqrt{2}}{(\Gamma(\frac{1}{4}))^2} \\ \\ \approx 0.67597824006728472899544768467080574828728345491540595197686291405\ldots$

$\xi(4) = \xi(-3) = \dfrac{\zeta(4)}{\zeta(2)} = -\dfrac{\pi^2}{3} \, \dfrac{B_4}{B_2} = \dfrac{\pi^2}{15} \\ \\ \approx 0.6579736267392905745889660666584100756875799604827193750942232917\ldots$

$\dfrac{\sqrt{e}}{\sqrt{2\pi}} \approx 0.657744623479456914069678728771474515096939699524536303950147\ldots$

$\dfrac{\zeta'(-3)}{\zeta(-3)} = -\dfrac{11}{6} - 120\,\ln A_3 \\ \\ \approx 0.6454291629329161373300369052496611474797327690754608471336427867\ldots$

$\zeta(2) - 1 = \dfrac{\pi^2}{6} \,-\, 1 \approx 0.64493406684822643647241516664602518921894990120679\ldots$

$\cot 1 \approx 0.64209261593433070300641998659426562023027811391817137910116\ldots$

$\dfrac{\xi'(2)}{\xi(2)} - \dfrac{\zeta'(2)}{\zeta(2)} = \dfrac{1}{2} H_1 + 1 - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{3}{2} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.63902722462453348262503027928226942865527292557238241484030484682\ldots$

$\dfrac{2}{\pi} \approx 0.636619772367581343075535053490057448137838582961825794990669376\ldots$

$\dfrac{\gamma+\ln 2}{2} \approx 0.635181422730739085011872105770289499558829735150089426463\ldots$

$\ln\!\left(\dfrac{3}{2}\sqrt{\dfrac{\pi}{2}}\right) = \dfrac{\ln\pi}{2} \,-\, \dfrac{3\ln 2}{2} \,+\, \ln 3 \approx 0.631256460752891814341110730411790208\ldots$

$\dfrac{\ln(2\pi)}{2}\,+\,\dfrac{\gamma}{2} \approx 0.6303307007539063114770736913644164243403178056678216134\ldots$

$\dfrac{\Gamma(\frac{1}{2})}{2\sqrt{2}} = \dfrac{\sqrt{\pi}}{2\sqrt{2}} = \sqrt{\dfrac{\pi}{8}} \approx 0.62665706865775012560394132120276131325174668515\ldots$

$-\ln(\cos1) \approx 0.6156264703860142621470375164088918633509354239463728341\ldots$

$H_\frac{1}{2} = H_{-\frac{1}{2}} + 2 - \pi\cot(\frac{\pi}{2}) = H_{-\frac{3}{2}} = 2(1-\ln 2) \\ \\ \approx 0.61370563888010938116553575708364686384899973127948949175863998101\ldots$

$\dfrac{\Gamma(-\frac{1}{3})}{\Gamma(-\frac{5}{6})} \approx 0.60817511033928118716015127601638137398369998997345464486511\ldots$

$\dfrac{\xi(4)}{\zeta(4)} = \dfrac{6}{\pi^2} \approx 0.60792710185402662866327677925836583342615264803347929307\ldots$

$\eta\!\left(\frac{1}{2}\right) \approx 0.604898643421630370247265914235955499759762545130247380378546\ldots$

$\dfrac{\pi}{3\sqrt{3}} \approx 0.604599788078072616864692752547385244094688749364246858523294\ldots$

$\dfrac{\pi}{2\varpi} = \dfrac{\pi\sqrt{2\pi}}{\left(\Gamma(\frac{1}{4})\right)^2} \approx 0.5990701173677961037199612461401619391136063316078257\ldots$

$G = -e \, \text{Ei}(-1) = e \, E_1(1) \approx 0.596347362323194074341078499369279376074177\ldots$

$\ln(\cosh(1)\sinh(1)) = \ln(\sinh2) - \ln 2 \\ \\ \approx 0.595220192054222820636614413342477235321634973066002079192846546\ldots$

$\dfrac{1}{e-1} = \dfrac{1}{2} \coth\left(\dfrac{1}{2}\right) \,-\, \dfrac{1}{2} \approx 0.5819767068693264243850020051090115585468693\ldots$

$\dfrac{\zeta'(4)}{\zeta(4)} + \dfrac{\zeta'(-3)}{\zeta(-3)} = -H_3 + \ln(2\pi) + \gamma = -\dfrac{11}{6} + \ln(2\pi) + \gamma \\ \\ \approx 0.58175939797754501083383822956030437743162094988215709110673698252\ldots$

$\dfrac{\zeta(8)}{\xi(8)} = \dfrac{\pi^4}{168} \approx 0.5798160180595383168835734088613399479150451528136037005\ldots$

$\dfrac{1}{\sqrt{3}} = \dfrac{\sqrt{3}}{3} \approx 0.57735026918962576450914878050195745564760175127012687601\ldots$

$\gamma = \zeta(1) = \dfrac{\zeta'(1)}{\zeta(1)} = \ln\Gamma(0) \\ \\ \approx 0.57721566490153286060651209008240243104215933593992359880576723488\ldots$

$\lambda_5 \approx 0.57554271446117745243110640549286383356745661517979953952924758\ldots$

$\dfrac{2}{\sqrt{3}} \, \xi\left(\frac{1}{3}\right) = -\dfrac{2\,\Gamma\left(\frac{1}{3}\right) \, \zeta\left(\frac{2}{3}\right)}{9\sqrt{3}\,\pi^{\frac{1}{3}}} \approx 0.57439415295525847528552020438108046769367\ldots$

$\xi(3) = \xi(-2) = \dfrac{3}{2\pi}\,\zeta(3) \approx 0.57393989404675551337522971036129920553897559\ldots$

$\dfrac{\ln\pi}{2} = \ln\!\sqrt{\pi} \approx 0.57236494292470008707171367567652935582364740645765578\ldots$

$-\dfrac{\zeta'(2)}{\zeta(2)} = - \ln(2\pi) - \gamma + 12\ln{A} \\ \\ \approx 0.56996099309453280639986436001973000240348228080693097955819736044\ldots$

$\dfrac{1}{\sqrt{\pi}} \approx 0.5641895835477562869480794515607725858440506293289988568440857\ldots$

$\dfrac{\xi'(5)}{\xi(5)} + \dfrac{\zeta'(-4)}{\zeta(-4)} = \dfrac{1}{2} H_\frac{5}{2} - H_3 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = -\dfrac{3}{10} + \dfrac{\ln\pi}{2} + \dfrac{\gamma}{2} \\ \\ \approx 0.56097277537546651737496972071773057134472707442761758515969515318\ldots$

$\dfrac{\xi'(0)}{\xi(0)} + \dfrac{\zeta'(1)}{\zeta(1)} = \dfrac{1}{2} H_0 - H_{-2} + \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} = -1 + \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.5541199559354118267922018421759071394202272087878728392803751626\ldots$

$-\ln{\gamma} \approx 0.54953931298164482233766176880290778833069898126306479109015\ldots$

$\dfrac{\sqrt{\pi}}{2} \, \text{erfi}\left(\dfrac{1}{2}\right) \approx 0.54498710418362222366242013140701667104079810362893518\ldots$

$2\pi \, \text{csch}(\pi) = \dfrac{2\pi}{\sinh(\pi)} \approx 0.5440581099642663259004731673440751116814367269\ldots$

$\dfrac{\Gamma(\frac{2}{3})}{\sqrt{2\pi}} = \dfrac{\sqrt{2\pi}}{\sqrt{3}\,\Gamma(\frac{1}{3})} \approx 0.54021489868725726519956542690723810040622912201765\ldots$

$1\,-\,\tanh\!\left(\dfrac{1}{2}\right) = \dfrac{2}{e+1} \approx 0.537882842739990241497681516356327451269710719\ldots$

$\dfrac{2\sqrt{2}}{3\sqrt{\pi}} = \sqrt{\dfrac{8}{9\pi}} \approx 0.53192304053524357058659474657917582463447817488657954\ldots$

$\xi(2) = \xi(-1) = \dfrac{\pi}{6} \approx 0.5235987755982988730771072305465838140328615665625\ldots$

$H_\frac{2}{5} = H_{-\frac{2}{5}} + \dfrac{5}{2} - \pi\cot(\frac{2\pi}{5}) = \dfrac{5}{2} - \dfrac{\pi}{2} \sqrt{1-\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} + \dfrac{\sqrt{5}}{4} \ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx 0.5158311203164167148758366080348739752157982394291134415718135596\ldots$

$\eta(0) = -\zeta(0) = \xi(0) = \xi(1) = \dfrac{1}{2} \tan\!\left(\dfrac{\pi}{4}\right) = \dfrac{1}{2} \cot\!\left(\dfrac{\pi}{4}\right) = 0.5$

$\xi\left(\frac{1}{3}\right) = -\dfrac{\Gamma\left(\frac{1}{3}\right) \, \zeta\left(\frac{2}{3}\right)}{9\,\pi^{\frac{1}{3}}} \approx 0.4974399282444983340815261557537025223890293711\ldots$

$\xi\left(\frac{1}{2}\right) = -\dfrac{\Gamma\left(\frac{1}{4}\right) \, \zeta\left(\frac{1}{2}\right)}{8\,\pi^{\frac{1}{4}}} \approx 0.497120778188314109912773739685397719807293609\ldots$

$\dfrac{\Gamma(-\frac{3}{4})}{2\,\Gamma(-\frac{1}{4})} = \dfrac{\Gamma(\frac{1}{4})}{6\,\Gamma(\frac{3}{4})} = \dfrac{\left(\Gamma(\frac{1}{4})\right)^2}{6\pi\sqrt{2}} \approx 0.4931125198647731487184702262879532610791\ldots$

$\dfrac{\sqrt{2}}{\sqrt{\pi\,e}} \approx 0.483941449038286699595660385871121309657343941474870050975110\ldots$

$-\dfrac{1}{2}\,+\,\ln 2\,+\,\dfrac{\gamma}{2} \approx 0.4817550130107117397204881664993777835965798023302170\ldots$

$(\ln 2)^2 \approx 0.4804530139182014246671025263266649717305529515945455868668\ldots$

$\dfrac{\xi(3)}{\zeta(3)} = \dfrac{3}{2\pi} \approx 0.4774648292756860073066512901175430861033789372213693462\ldots$

$\ln\!\left(\!\sqrt{\dfrac{8}{\pi}}\right) = \dfrac{3\ln 2}{2} \,-\, \dfrac{\ln\pi}{2} \approx 0.46735582791521787705413450651073549628960\ldots$

$1\,-\,\sqrt{2}\,+\,\ln(1+\sqrt{2}) \approx 0.46716002464644797643092060077009423045848845288\ldots$

$\tanh\!\left(\dfrac{1}{2}\right) = \dfrac{e-1}{e+1} \approx 0.462117157260009758502318483643672548730289280330\ldots$

$\dfrac{2\pi^2}{\cosh(\pi\sqrt{2}) - \cos(\pi\sqrt{2})} \approx 0.46139068323848162334195576672452318794112891\ldots$

$1\,-\,\cos1 \approx 0.459697694131860282599063392557023396267689579382077772329\ldots$

$\ln(e-1) \,-\, 1 \approx 0.458675145387081891021643645067329701876977906692194144\ldots$

$P_2 \approx 0.45224742004106549850654336483224793417323134323989242173641893\ldots$

$\ln\!\left(\dfrac{\pi}{2}\right) \approx 0.45158270528945486472619522989488214357179467855505631739294\ldots$

$\dfrac{1}{6}\,\Gamma(\frac{1}{3}) \approx 0.44648975578462460560928215682911294068811489632621685015840\ldots$

$H_\frac{1}{3} = H_{-\frac{1}{3}} + 3 - \pi\cot(\frac{\pi}{3}) = 3 - \dfrac{\pi}{2\sqrt{3}} - \dfrac{3\ln 3}{2} \\ \\ \approx 0.44518188488072653761009301579513357688673103921950553461301603184\ldots$

$\dfrac{\sqrt{e}}{e+1} \approx 0.443409441985036954329448898891704281267044544356306962418129\ldots$

$\dfrac{\gamma}{3} \,-\, \dfrac{\pi}{6\sqrt{3}} \,+\, \dfrac{\ln 3}{2} \approx 0.439411471928862824134113605548371040623787349542559\ldots$

$\ln(\cosh1) \approx 0.43378083048302718702649468490012786335883292844810310342\ldots$

$\dfrac{\gamma}{2} \,+\, \ln 2 \,-\, \dfrac{\ln 3}{2} \approx 0.4324488686766568940228655480381149312728345234188423\ldots$

$-\dfrac{1}{6} \, H_{-\frac{2}{3}} = \dfrac{\pi}{12\sqrt{3}} \,+\, \dfrac{\ln 3}{4} \approx 0.42580301918654557706498449736747773718554482\ldots$

$\Psi^{(0)}(2) = \dfrac{\Gamma'(2)}{\Gamma(2)} = \Gamma'(2) = H_1 - \gamma = \Psi^{(0)}(-1) = \dfrac{\Gamma'(-1)}{\Gamma(-1)} = \\ \\ H_{-2} \,-\, \gamma = \! 1 \,-\, \gamma \approx 0.42278433509846713939348790991759756895784066406007\ldots$

$1\,-\,\dfrac{1}{e\!-\!1} \approx 0.418023293130673575614997994890988441453130698924603863733\ldots$

$\dfrac{\zeta(9)}{\xi(9)} = \dfrac{4\pi^4}{945} \approx 0.412313612842338358672763312968063962961809886445229298\ldots$

$\pi \, \text{sech}\left(\dfrac{\pi\sqrt{3}}{2}\right) \approx 0.411829422582171159051691874872279022515826006340363\ldots$

$1 - G = 1 - e\text{Ei}(-1) = 1 - eE_1(1) \\ \approx 0.40365263767680592565892150063072062392582213984745121842651508952\ldots$

$\dfrac{\zeta(4)}{\zeta(2)^2} = 0.4$

$\dfrac{1}{\sqrt{2\pi}} \approx 0.398942280401432677939946059934381868475858631164934657665925\ldots$

$\dfrac{\xi'(10)}{\xi(10)} - \dfrac{\zeta'(10)}{\zeta(10)} = \dfrac{1}{2} H_5 + \dfrac{1}{9} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{451}{360} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.3918050024023112604028080570600472064330507033501601926180826246\ldots$

$\ln\left(\dfrac{\Gamma^2(\frac{1}{4})}{2\pi\sqrt{2}}\right) = 2\ln\!\left(\dfrac{\Gamma(\frac{1}{4})}{2^\frac{1}{4}\sqrt{2\pi}}\right) = -\ln(2\pi) - \dfrac{\ln 2}{2} + 2\ln\Gamma(\frac{1}{4}) \\ \\ \approx 0.3915943927068367764719453468991110280902101157700266483053309593\ldots$

$\dfrac{\xi'(10)}{\xi(10)} = \dfrac{451}{360} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} - 132 \ln A_9 \\ \\ \approx 0.39110866195702723996621278036910991369639302684095675617166226960\ldots$

$\dfrac{\xi'(6)}{\xi(6)} + \dfrac{\zeta'(-5)}{\zeta(-5)} = \dfrac{1}{2} H_3 - H_4 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = -\dfrac{7}{6} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} \\ \\ \approx 0.38745328926874516012553517550924047275356054212120617261370849601\ldots$

$\dfrac{1}{2}\left(\theta_3\left(0,\dfrac{1}{e}\right)-1\right) \approx 0.386318602413326076515625275578929240671693022686\ldots$

$2\ln 2\,-\,1 \approx 0.38629436111989061883446424291635313615100026872051050824\ldots$

$\dfrac{\sqrt{\pi}-1}{2} \approx 0.386226925452758013649083741670572591398774728061193564106\ldots$

$\dfrac{\pi}{3\sqrt{3}} \,-\, \dfrac{\ln 2}{3} \approx 0.3735507278914241803922820453946597214028553712441617738\ldots$

$1\,+\,\dfrac{\gamma}{2}\,-\,\dfrac{\ln(2\pi)}{2} \approx 0.3696692992460936885229263086355835756596821943321783\ldots$

$\lambda_4 \approx 0.36879047949224163859051148963775607226216669396085280482311885\ldots$

$\dfrac{1}{e} \approx 0.367879441171442321595523770161460867445811131031767834507836801\ldots$

$\dfrac{\xi'(9)}{\xi(9)} - \dfrac{\zeta'(9)}{\zeta(9)} = \dfrac{1}{2} H_\frac{9}{2} + \dfrac{1}{8} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{4819}{2520} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.3581816313661754747950997451256801621670743785137144623069264246\ldots$

$H_\frac{1}{4} = H_{-\frac{1}{4}} + 4 - \pi\cot(\frac{\pi}{4}) = 4 - \dfrac{\pi}{2} - 3\ln 2 \\ \\ \approx 0.3497621315252674525169819439857188536749148972316813271504876753\ldots$

$\dfrac{\sqrt{2-\sqrt{2}}}{8}\,\Gamma(\frac{1}{4}) \approx 0.346865211023809496042035100047113325318118780571620\ldots$

$\dfrac{\ln 2}{2} = \ln\sqrt{2} \approx 0.346573590279972654708616060729088284037750067180127627\ldots$

$-\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{2\pi}\right) = \gamma - H_\frac{1}{2\pi} \\ \\ \approx 0.3420769869321474450726809742612898230134971513622385706894293845\ldots$

$\dfrac{\Gamma(\frac{3}{4})}{\Gamma(\frac{1}{4})} \approx 0.337989120033642364497723842335402874143641727457702975988431\ldots$

$\zeta'(1) = \gamma^2 \approx 0.33317792380771867431837613635524422665941714024962974315\ldots$

$\dfrac{\zeta'(5)}{\zeta(5)} + \dfrac{\zeta'(-4)}{\zeta(-4)} = -H_4 + \ln(2\pi) + \gamma = -\dfrac{25}{12} + \ln(2\pi) + \gamma \\ \\ \approx 0.33175939797754501083383822956030437743162094988215709110673698252\ldots$

$\gamma \, e^{-\gamma} \approx 0.324083209122330886046504542274496209475461672770485826786147\ldots$

$\dfrac{\xi'(8)}{\xi(8)} - \dfrac{\zeta'(8)}{\zeta(8)} = \dfrac{1}{2} H_4 + \dfrac{1}{7} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{199}{168} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.3235510341483430064345540888060789524647967350961919386498286563\ldots$

$\dfrac{1}{2}\,\ln\dfrac{6}{\pi} \approx 0.3235147916893274133345250035138217805378479396338465671708\ldots$

$\dfrac{\cot 1}{2} \approx 0.32104630796716535150320999329713281011513905695908568955058\ldots$

$\dfrac{\xi'(8)}{\xi(8)} = \dfrac{199}{168} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} + 240 \ln A_7 \\ \\ \approx 0.32066086584029625005100451395866035294358692847663990241977450633\ldots$

$\dfrac{\xi(2)}{\zeta(2)} = \dfrac{1}{\pi} \approx 0.31830988618379067153776752674502872406891929148091289749\ldots$

$\gamma \,-\, M_1 \approx 0.3157184520538900768510852514737065719905926876787243926137\ldots$

$1 \,-\, \ln 2 \approx 0.3068528194400546905827678785418234319244998656397447458793\ldots$

$\dfrac{\ln(2\pi)}{6} \approx 0.30631284440155758059344324546853921328713249121259447093905\ldots$

$2\sqrt{\pi}\,e^{-(\frac{\pi}{2})^2} \approx 0.30062580006472378026897076569351276802293029155385829030\ldots$

$\dfrac{\xi'(7)}{\xi(7)} - \dfrac{\zeta'(7)}{\zeta(7)} = \dfrac{1}{2} H_\frac{7}{2} + \dfrac{1}{6} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{129}{70} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.28873718692173103035065530068123571772262993406927001786248198019\ldots$

$-\ln\zeta(0) = \Gamma(0) \; \zeta(0) = \dfrac{\gamma}{2} \approx 0.288607832450766430303256045041201215521079\ldots$

$H_\frac{1}{5} = H_{-\frac{1}{5}} + 5 - \pi\cot(\frac{\pi}{5}) = 5 - \dfrac{\pi}{2}\sqrt{1+\dfrac{2}{\sqrt{5}}} - \dfrac{5}{4}\ln 5 - \dfrac{\sqrt{5}}{4}\ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx 0.28817576830934456505930412763245032621657106173328078862990857069\ldots$

$\dfrac{\sqrt{5}-1}{20}\,\Gamma(\frac{1}{5}) \approx 0.28372974510539937507390373117186467203932607707016463\ldots$

$\dfrac{\zeta(10)}{\xi(10)} = \dfrac{\pi^5}{1080} \approx 0.2833515599863717159840197315216996356299080616926620\ldots$

$\Psi^{(0)}(\frac{7}{8}) - \Psi^{(0)}(\frac{3}{4})= \dfrac{\Gamma'(\frac{7}{8})}{\Gamma(\frac{7}{8})} - \dfrac{\Gamma'(\frac{3}{4})}{\Gamma(\frac{3}{4})} = \dfrac{\pi}{\sqrt{2}} - \ln 2 - \dfrac{\ln(3+2\sqrt{2})}{\sqrt{2}} \\ \\ \approx 0.2818438082387767873026682130710342759304200856344553516245882417\ldots$

$\dfrac{\xi'(3)}{\xi(3)} - \dfrac{\zeta'(3)}{\zeta(3)} = \dfrac{1}{2} H_\frac{3}{2} + \dfrac{1}{2} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{11}{6} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.27921337739792150654113149115742619391310612454546049405295817066\ldots$

$\pi \, \text{csch}(\pi) = \dfrac{\pi}{\sinh(\pi)} \approx 0.27202905498213316295023658367203755584071836346\ldots$

$\dfrac{\gamma}{\zeta(\frac{1}{2})^2} \approx 0.270658404289314085591987051240908351797918940210368105827833\ldots$

$\dfrac{1}{2}\,-\,\dfrac{\ln(2\pi)}{2}\,+\,\dfrac{\gamma}{3}\,+\,2\ln{A} \approx 0.27097564249674007018301361410741112268072839\ldots$

$\Gamma(-\frac{7}{2}) = \dfrac{16\sqrt{\pi}}{105} \approx 0.270088205852269108921625521271031646902483726647220\ldots$

$2\,-\,\dfrac{5\ln2}{2} \approx 0.26713204860013672645691969635455857981124966409936186469\ldots$

$\Gamma(-3)\,\sin(2\pi) = \dfrac{\pi}{12} \approx 0.2617993877991494365385536152732919070164307832\ldots$

$M_1 \approx 0.26149721284764278375542683860869585905156664826119920619206421\ldots$

$\dfrac{2}{\pi^2} (H_\frac{\pi}{2} + H_{-\frac{\pi}{2}}) \approx 0.25976967104758191318248448162818198758030018555185330\ldots$

$2\zeta(3) \,-\, \dfrac{\pi^2}{6} \,-\, \dfrac{1}{2} \approx 0.2591797394709621343270611563768747923110226834741993\ldots$

$\dfrac{\xi'(6)}{\xi(6)} - \dfrac{\zeta'(6)}{\zeta(6)} = \dfrac{1}{2} H_3 + \dfrac{1}{5} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{67}{60} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.25569389129120014929169694594893609532193959223904908150697151349\ldots$

$\dfrac{\xi'(7)}{\xi(7)} + \dfrac{\zeta'(-6)}{\zeta(-6)} = \dfrac{1}{2} H_\frac{7}{2} - H_5 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = -\dfrac{17}{28} + \dfrac{\ln\pi}{2} + \dfrac{\gamma}{2} \\ \\ \approx 0.25382991823260937451782686357487342848758421728476044230255229604\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi^2}{6} = \ln\!\dfrac{\pi}{\sqrt{6}} \approx 0.248850151235372673737188672162707575285799466823809\ldots$

$\ln A \approx 0.2487544770337842625472529935761139760973697136685351169998556\ldots$

$-\dfrac{1}{3} \, H_{-\frac{1}{3}} = -\dfrac{\pi}{6\sqrt{3}} \,+\, \dfrac{\ln 3}{2} \approx 0.2470062502950185372652762421875702302764009\ldots$

$-1 \,+\, e(1-\ln(e-1)) \approx 0.24680831287151537038013735601030553525049783631\ldots$

$\dfrac{\xi'(6)}{\xi(6)} = \dfrac{67}{60} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} - 252 \ln A_5 \\ \\ \approx 0.24306082225868908846780465000310409541355574473892682572286894199\ldots$

$\dfrac{1}{\sqrt{2\pi\,e}} \approx 0.24197072451914334979783019293556065482867197073743502548755\ldots$

$\dfrac{(\ln 2)^2}{2} \approx 0.24022650695910071233355126316333248586527647579727279343343\ldots$

$\xi'(6) = \xi(6) \left(\dfrac{67}{60} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} - 252 \ln A_5\right) \!=\! \dfrac{2\pi^3}{63} \left(\dfrac{67}{60} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} - 252 \ln A_5\right) \\ \\ \approx 0.23925114619346320474637351872462488568238885282754833756330525416\ldots$

$2\gamma\,-\,\dfrac{\ln(2\pi)}{2} \approx 0.235492796598392979432694443759187222222921198242063784\ldots$

$H_\frac{1}{2\pi} \approx 0.2351386779693854155338311158211126080286621845776850281163378\ldots$

$\dfrac{\xi'(5)}{\xi(5)} - \dfrac{\zeta'(5)}{\zeta(5)} = \dfrac{1}{2} H_\frac{5}{2} + \dfrac{1}{4} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{107}{60} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.22921337739792150654113149115742619391310612454546049405295817066\ldots$

$\dfrac{1}{2} \ln\!\left(\dfrac{\pi}{2}\right) \approx 0.225791352644727432363097614947441071785897339277528158696\ldots$

$\dfrac{\xi'(4)}{\xi(4)} - \dfrac{\zeta'(4)}{\zeta(4)} = \dfrac{1}{2} H_2 + \dfrac{1}{3} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = \dfrac{13}{12} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} \\ \\ \approx 0.22236055795786681595836361261560276198860625890571574817363818015\ldots$

$-\text{Ei}(-1) = E_1(1) = \dfrac{G}{e} \approx 0.21938393439552027367716377546012164903104729\ldots$

$\dfrac{1}{e(e\!-\!1)} \approx 0.2140972656978841027894782349475506911010581700436283017589\ldots$

$\dfrac{1-\zeta(1)}{2} = \dfrac{1-\gamma}{2} \approx 0.2113921675492335696967439549587987844789203320300\ldots$

$\dfrac{\gamma}{3} \,+\, \dfrac{\pi}{6\sqrt{3}} \,+\, \dfrac{\ln 3}{2} \,-\, \dfrac{\ln 2}{3} \approx 0.2083624117422143876617028983956455179319539714\ldots$

$\dfrac{\zeta(\frac{3}{2})}{4\pi} = -\zeta(-\frac{1}{2}) \approx 0.207886224977354566017306725397049302226268531287672\ldots$

$\lambda_3 \approx 0.20763892055432480379149204661780320698263607917960073085244812\ldots$

$\zeta(3)-1 \approx 0.2020569031595942853997381615114499907649862923404988817922\ldots$

$-\zeta'(3) \approx 0.19812624288563685333068182150328579687554279346383500334688\ldots$

$\ln 2 \,-\, \dfrac{1}{2} \approx 0.193147180559945309417232121458176568075500134360255254120\ldots$

$\dfrac{1}{2e} \approx 0.18393972058572116079776188508073043372290556551588391725391840\ldots$

$1\,-\,\dfrac{\zeta(2)}{2} = 1\,-\,\dfrac{\pi^2}{12} \approx 0.177532966575886781763792416676987405390525049396\ldots$

$\dfrac{\gamma}{2} \,+\, \ln 2 \,-\, \dfrac{\ln 5}{2} \approx 0.1770360567936615524201084998862839638337791251959581\ldots$

$-\ln(\sin1) \approx 0.1726037462690916785134109758639090698401084088964048049\ldots$

$-\dfrac{1}{2} \,+\, e(-1+e(1-\ln(e-1))) \\ \\ \approx 0.17089455199127491019799706794233736593767978819561354848800982279\ldots$

$\dfrac{\zeta'(1)}{2} = \dfrac{\gamma^2}{2} \approx 0.1665889619038593371591880681776221133297085701248148715\ldots$

$-\zeta^\prime(-1) = \ln A \,-\, \dfrac{1}{12} \approx 0.1654211437004509292139196602427806427640363803\ldots$

$-\ln\!\left(\dfrac{8}{3\pi}\right) = -3\ln 2 \,+\, \ln\pi \,+\, \ln 3 \approx 0.163900632837673937286976223901054712\ldots$

$\dfrac{\cot 1}{4} \approx 0.16052315398358267575160499664856640505756952847954284477529\ldots$

$\dfrac{1}{2\pi} \approx 0.15915494309189533576888376337251436203445964574045644874766734\ldots$

$\dfrac{\xi'(4)}{\xi(4)} = \dfrac{13}{12} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} + 120 \ln A_3 \\ \\ \approx 0.15869079300249568946216493692624599194049443971241199214673237587\ldots$

$\text{erfc}\,(1) = 1 \,-\, \text{erf}\,(1) \approx 0.15729920705028513065877936491739074070393300203\ldots$

$\sqrt{\pi}\,e^{-(\frac{\pi}{2})^2} \approx 0.150312900032361890134485382846756384011465145776929145153\ldots$

$-\dfrac{1}{\sqrt{2}} \, \zeta(-\frac{1}{2}) \approx 0.146997759396759645698712220487214416199301425169869947\ldots$

$\dfrac{\xi'(8)}{\xi(8)} + \dfrac{\zeta'(-7)}{\zeta(-7)} = \dfrac{1}{2} H_4 - H_6 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = -\dfrac{169}{120} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} \\ \\ \approx 0.14578662260207849345886850884257380608689387545453950594704182934\ldots$

$\dfrac{\pi^2}{6} \,-\, \dfrac{3}{2} \approx 0.1449340668482264364724151666460251892189499012067984377355\ldots$

$\dfrac{\zeta'(-5)}{\zeta(-5)} = -\dfrac{137}{60} + 252\,\ln A_5 \\ \\ \approx 0.1443924670100560716577305255061363773400047973822793468908395540\ldots$

$\dfrac{\sqrt{\pi}}{2} \, \text{erfc}\,(1) = \dfrac{\sqrt{\pi}}{2} \,(1\,-\,\text{erf}\,(1)) \approx 0.13940279264033098824961630553871958604\ldots$

$\dfrac{155925\;\zeta(11)}{4\pi^{11}} \approx 0.13256281935954208462677109744716735379524560552128793\ldots$

$\dfrac{\zeta'(6)}{\zeta(6)} + \dfrac{\zeta'(-5)}{\zeta(-5)} = -H_5 + \ln(2\pi) + \gamma = -\dfrac{137}{60} + \ln(2\pi) + \gamma \\ \\ \approx 0.13175939797754501083383822956030437743162094988215709110673698252\ldots$

$-\dfrac{1}{3} + e\left(-\dfrac{1}{2}+e(-1+e(1-\ln(e-1)))\right) \\ \\ \approx 0.13120622192719879945083965355792969194268624574062097742819030468\ldots$

$-\dfrac{1}{6} \, H_{-\frac{1}{3}} = -\dfrac{\pi}{12\sqrt{3}} \,+\, \dfrac{\ln 3}{4} \approx 0.1235031251475092686326381210937851151382004\ldots$

$\ln\!\left(\dfrac{2}{\sqrt{\pi}}\right) = \ln 2 \,-\, \dfrac{\ln\pi}{2} \approx 0.120782237635245222345518445781647212251852727\ldots$

$\dfrac{1}{\pi\,e} \approx 0.11709966304863832138048453693333374442782984255212289775394452\ldots$

$\dfrac{\xi'(3)}{\xi(3)} \approx 0.114390695239644266354638096357951128890096544123057223181212\ldots$

$\dfrac{\ln 2}{2\pi} \approx 0.1103178000763257966982282160589988454913448743648277352446316\ldots$

$-\dfrac{1}{4} + e\left(-\dfrac{1}{3} + e\left(-\dfrac{1}{2}+e(-1+e(1-\ln(e-1)))\right)\right) \\ \\ \approx 0.10665548884546922651593889331785110213877727951994160011253493342\ldots$

$\xi'(4) = \xi(4) \left(\dfrac{13}{12} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} + 120 \ln A_3\right) = \dfrac{\pi^2}{15} \left(\dfrac{13}{12} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} + 120 \ln A_3\right) \\ \\ \approx 0.10441435660198612338558436722401686882411630778947897015484634166\ldots$

$\dfrac{\zeta(3)\!-\!1}{2} \approx 0.101028451579797142699869080755724995382493146170249440896\ldots$

$\dfrac{1}{2}\,-\,\dfrac{\zeta(3)}{3} \approx 0.099314365613468571533420612829516669745004569219833706069\ldots$

$\lambda_2 = 1 + \dfrac{\pi^2}{8} - 2\ln 2 - \ln\pi + \gamma(1\!-\!\gamma) - 2\gamma_1 \\ \\ \approx 0.09234573522804667038572848619206788677413221662824650939963259793\ldots$

$\dfrac{\gamma}{2\pi} \approx 0.09186672629915399037964223407187809141362928056064121236108720\ldots$

$-\dfrac{1}{3}\,+\,\dfrac{\ln(2\pi)}{2}\,-\,2\ln A \approx 0.0880962458037708833524904159200563543333247129\ldots$

$\dfrac{\gamma}{2} \,+\, \dfrac{\ln 2}{2} \,-\, \dfrac{\ln 3}{2} \approx 0.0858752783966842393142494873090266472350844562387147\ldots$

$1 \,-\, \dfrac{1}{2} \cot\dfrac{1}{2} \approx 0.0847561391437740403659902805155916881209460259919329978\ldots$

$\dfrac{1}{e-1} \,-\, \dfrac{1}{2} = \dfrac{1}{2} \coth\left(\dfrac{1}{2}\right) \,-\, 1 \approx 0.0819767068693264243850020051090115585468\ldots$

$-\dfrac{1}{2}\,\ln\!\left(\dfrac{8}{3\pi}\right) = \dfrac{1}{2}\left(-3\ln 2 + \ln\pi + \ln 3\right) \approx 0.0819503164188369686434881119505\ldots$

$1\,+\,\zeta'(0) = 1\,-\,\dfrac{\ln(2\pi)}{2} \approx 0.081061466795327258219670263594382360138602526\ldots$

$\dfrac{1}{2}\,-\,\dfrac{\ln(2\pi)}{2}\,+\,2\ln A \approx 0.078570420862895783314176250746610312333341953699\ldots$

$\gamma\,-\,\dfrac{1}{2} \approx 0.07721566490153286060651209008240243104215933593992359880576\ldots$

$\dfrac{\xi'(2)}{\xi(2)} = -\dfrac{\xi'(-1)}{\xi(-1)} = \dfrac{3}{2} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} - 12\ln{A} \\ \\ \approx 0.06906623153000067622516591926253942625179064476545143528210748638\ldots$

$\xi'(3) = \dfrac{3}{2\pi} \, \left(\zeta'(3) + \zeta(3)\left(\dfrac{11}{6} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \right)\right) \\ \\ \approx 0.06565338350577613051056776848576336697546994559255567505515985927\ldots$

$\dfrac{1}{3}\,-\,\dfrac{\zeta(4)}{4} = \dfrac{1}{3}\left(1\,-\,\dfrac{\pi^4}{120}\right) \approx 0.062752524905548785454332409198041357639645\ldots$

$\zeta'(2) \,+\, 1 = 1 \,+\, \zeta(2) \left(\gamma + \ln{(2\pi)} - 12\ln{A}\right) = 1 \,+\, \dfrac{\pi^2}{6} \left(\gamma + \ln{(2\pi)} - 12\ln{A}\right) \\ \\ \approx 0.0624517456841562462974259054321350221021397113851700741145665196\ldots$

$\dfrac{3\;\zeta(3)}{2\pi^{3}} \approx 0.05815226940437519841167983547808172470935185205932886883469\ldots$

$-\dfrac{1}{2\pi}\,\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{2\pi}\right) = \dfrac{\gamma - H_\frac{1}{2\pi}}{2\pi} \\ \\ \approx 0.05444324338823295106107592412560811015807868373898649441210240548\ldots$

$\dfrac{1}{3}\,-\,\dfrac{\gamma}{2} \approx 0.04472550088256690303007728829213211781225366536337153393044\ldots$

$\dfrac{\theta_3(0,e^{-\pi})-1}{2} \approx 0.04321740560665400728765806075511172853510285362260944\ldots$

$\dfrac{1}{4}\,-\,\dfrac{\zeta(5)}{5} \approx 0.042614448971326014733726902708593166388583816099617437605\ldots$

$\dfrac{1}{2} \,-\, \dfrac{1}{4} \cot\dfrac{1}{2} \approx 0.0423780695718870201829951402577958440604730129959664989\ldots$

$-\dfrac{\ln 2}{2}\,-\,\dfrac{\ln(1\!-\!\cos 1)}{2} \approx 0.04201950582536896172579838403790203712453892055\ldots$

$\dfrac{\zeta(2)}{4\pi^2} = \dfrac{1}{24} \approx 0.04166666666666666666666666666666666666666666666666666666\ldots$

$2\,-\,\dfrac{\ln\pi}{2}\,-\,2\ln 2 \approx 0.041340695955409294093822081407117508025352324821833\ldots$

$\dfrac{1}{4}\coth\dfrac{1}{2} \,-\, \dfrac{1}{2} = \dfrac{3-e}{4(e-1)} \approx 0.04098835343466321219250100255450577927343465\ldots$

$\dfrac{1+\zeta'(0)}{2} = \dfrac{1}{2}\,-\,\dfrac{\ln(2\pi)}{4} \approx 0.040530733397663629109835131797191180069301263\ldots$

$\dfrac{\gamma}{2}\,-\,\dfrac{1}{4} \approx 0.03860783245076643030325604504120121552107966796996179940288\ldots$

$\Psi^{(0)}(\frac{3}{2}) = \dfrac{\Gamma'(\frac{3}{2})}{\Gamma(\frac{3}{2})} = H_{\frac{1}{2}} - \gamma = \Psi^{(0)}(-\frac{1}{2}) = \dfrac{\Gamma'(-\frac{1}{2})}{\Gamma(-\frac{1}{2})} = H_{-\frac{3}{2}} - \gamma \\ \\ = 2\,-\,2\ln 2\,-\,\gamma \approx 0.036489973978576520559023667001244432806840395339565\ldots$

$\dfrac{\pi^2}{24} - \dfrac{3}{8} \approx 0.03623351671205660911810379166150629730473747530169960943388\ldots$

$\xi'(2) = -\xi'(-1) = \dfrac{\pi}{6} \left(\dfrac{3}{2} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} - 12\ln{A}\right) \\ \\ \approx 0.0361629942642969783127504756099367089998206029232772312349852718\ldots$

$-H_{-\frac{\pi}{2}} \approx 0.03259686937614189516312738735001334130268817312652775679552\ldots$

$\ln A_2 = \dfrac{\zeta(3)}{4\pi^2} \approx 0.03044845705839327078025153047115477664700048354497393\ldots$

$\dfrac{1}{5}\,-\,\dfrac{\zeta(6)}{6} = \dfrac{1}{5}\left(1\,-\,\dfrac{\pi^6}{1134}\right) \approx 0.03044282300259181004758034503484657868303\ldots$

$\dfrac{3}{4}\,-\,\dfrac{1}{2\ln 2} \approx 0.02865247955551829632003765949905393128667702292350703293\ldots$

$3\,-\,\dfrac{\ln\pi}{2}\,+\,\dfrac{\ln 2}{2}\,-\,\dfrac{5\ln 3}{2} \approx 0.027677925684998339148789292746244666595376266\ldots$

$\dfrac{315\;\zeta(7)}{4\pi^{7}} \approx 0.026291323260780709375874073252262584179775839233514745808\ldots$

$-\zeta(-\frac{3}{2}) = \dfrac{3}{16\pi^2} \, \zeta(\frac{5}{2}) \approx 0.02548520188983303594954298691070474546902498460\ldots$

$1 - \ln(e\!-\!1) - \dfrac{1}{e} - \dfrac{1}{2e^2} \\ \\ \approx 0.02312806259733322347912012741962663272735100270563836959308192776\ldots$

$\lambda_1 = \dfrac{\xi'(1)}{\xi(1)} = -\dfrac{\xi'(0)}{\xi(0)} = 2\,\xi'(1) = -2\,\xi'(0) = 1 - \ln 2 - \dfrac{\ln\pi}{2} + \dfrac{\gamma}{2} \\ \\ \approx 0.02309570896612103381431024790649529162193212715205075952539207221\ldots$

$\dfrac{1}{6}\,-\,\dfrac{\zeta(7)}{7} \approx 0.022616769897820548546695588116695701009543300871632565749\ldots$

$\dfrac{\zeta(2)}{8\,\pi^2} = \dfrac{\eta(2)}{4\,\pi^2} = \dfrac{1}{48} \approx 0.0208333333333333333333333333333333333333333333333\ldots$

$4\,-\,\dfrac{\ln\pi}{2}\,-\,\dfrac{13\ln 2}{2}\,+\,\ln 3 \approx 0.02079067210376509311152277176784865633309227\ldots$

$\dfrac{467775\,\zeta(13)}{8\pi^{13}} \approx 0.02013965218220312979358418257542589230524569195444318\ldots$

$-\dfrac{3\,\zeta(3)}{2\pi^4} \approx 0.018510442255435801952924479643569175592265319064455292533\ldots$

$\dfrac{1}{2} \, \Psi^{(0)}(-\frac{1}{2}) = 1 - \ln 2 - \dfrac{\gamma}{2} \approx 0.01824498698928826027951183350062221640342019\ldots$

$\dfrac{1}{7}\,-\,\dfrac{\zeta(8)}{8} = \dfrac{1}{7}\left(1\,-\,\dfrac{\pi^8}{10800}\right) \approx 0.0173474733323998147205214880435612989854\ldots$

$5\,-\,\dfrac{\ln\pi}{2}\,+\,\dfrac{5\ln 2}{2}\,+\,\ln 3\,-\,\dfrac{9\ln 5}{2} \approx 0.01664469118982119216319486537359339114\ldots$

$6\,-\,\dfrac{\ln\pi}{2}\,-\,3\ln 2\,-\,\dfrac{9\ln 3}{2}\,+\,\ln 5 \approx 0.013876128823070747998745727023762908561\ldots$

$\dfrac{1}{8}\,-\,\dfrac{\zeta(9)}{9} \approx 0.013665734130435309509127470085287548834932683178345693716\ldots$

$7\,-\,\dfrac{\ln\pi}{2}\,+\,\dfrac{7\ln 2}{2}\,+\,2\ln 3\,+\,\ln 5\,-\,\dfrac{13\ln 7}{2} \approx 0.0118967099458917700950557241176\ldots$

$\xi'(1) = \dfrac{1}{2} \,-\, \dfrac{\ln(4\pi)}{4} \,+\, \dfrac{\gamma}{4} \approx 0.01154785448306051690715512395324764581096606\ldots$

$8\,-\,\dfrac{\ln\pi}{2}\,-19\ln 2\,+\,2\ln 3\,+\,\ln 5\,+\,\ln 7 \approx 0.010411265261972096497478567132534\ldots$

$\dfrac{\zeta(3)}{4\pi^3} \approx 0.009692044900729199735279972579680287451558642009888144805781\ldots$

$9\,-\,\dfrac{\ln\pi}{2}\,+\,\dfrac{13\ln 2}{2}\,-\,15\ln 3\,+\,\ln 5\,+\,\ln 7 \approx 0.00925546218271273291772863663310\ldots$

$\dfrac{\gamma}{2} \,+\, \ln 2 \,-\, \dfrac{\ln 7}{2} \approx 0.0087999384830550871678117947777879187780374375392864\ldots$

$\zeta(-\frac{5}{2}) = \dfrac{15}{64\pi^3} \, \zeta(\frac{7}{2}) \approx 0.008516928777850330542358567028344486936275990220\ldots$

$\zeta(-3) = \dfrac{1}{120} \approx 0.00833333333333333333333333333333333333333333333333333\ldots$

$10\,-\,\dfrac{\ln\pi}{2}\,-\,3\ln 2\,+\,4\ln 3\,-\,\dfrac{17\ln 5}{2}\,+\,\ln 7 \approx 0.008330563433362871256469318659\ldots$

$\zeta'(-8) = \dfrac{315\,\zeta(9)}{4\pi^8} = -\ln A_8 \approx 0.008316161985602247359524426510534214225\ldots$

$\zeta'(-4) = \dfrac{3\,\zeta(5)}{4\pi^4} = -\ln A_4 \approx 0.00798381145026862428069667079878930390523\ldots$

$\dfrac{3\,\zeta(3)}{16\,\pi^3} = \dfrac{\eta(3)}{4\,\pi^3} \approx 0.0072690336755468998014599794347602155886689815074161\ldots$

$1 - \ln(e\!-\!1) - \dfrac{1}{e} - \dfrac{1}{2e^2} - \dfrac{1}{3e^3} \\ \\ \approx 0.00653237314137857581933932220293937385011780530949729773720601856\ldots$

$\text{Li}_{-3}\left(\frac{1}{e}\right) - \Gamma(4) = \dfrac{e(e^2+4e+1)}{(e-1)^4} \,-\, 6 \\ \\ \approx 0.00651279663676014827329730289997835030570102171240478259383392322\ldots$

$\dfrac{14175\,\zeta(11)}{8\pi^{11}} \approx 0.006025582698161003846671413520325788808874800250967633\ldots$

$\dfrac{14501025\,\zeta(11)}{8192\,\pi^{11}} = \dfrac{14175\,\eta(11)}{8\,\pi^{11}} \approx 0.00601969834005733099135239846805984565\ldots$

$\ln A_6 = \dfrac{45}{8\pi^6} \, \zeta(7) \approx 0.00589975914351593745062987740839202557980153462015\ldots$

$\zeta'(-3) = -\dfrac{11}{720} \,-\, \ln A_3 \approx 0.00537857635777430114441697421041384289566443\ldots$

$\text{Li}_{-8}\left(\frac{1}{e}\right) - \Gamma(9) = \dfrac{e(e^7+247e^6+4293e^5+15619e^4+15619e^3+4293e^2+247e+1)}{(e-1)^9} \\ \\ - \, 40320 \approx 0.004674583314077467976720610355903775815702898024620161911\ldots$

$\zeta(-\frac{7}{2}) = \dfrac{105}{256\pi^4} \, \zeta(\frac{9}{2}) \approx 0.00444101133547943195853465801781977508621424544\ldots$

$\zeta(-7) = \dfrac{1}{240} \approx 0.00416666666666666666666666666666666666666666666666666\ldots$

$\dfrac{2825\,\zeta(10)}{8\pi^{10}} = \dfrac{1}{264} \approx 0.0037878787878787878787878787878787878787878787878\ldots$

$\dfrac{1448685\,\zeta(10)}{4096\,\pi^{10}} = \dfrac{2835\,\eta(10)}{8\,\pi^{10}} = \dfrac{511}{135168} \approx 0.003780480587121212121212121212\ldots$

$\dfrac{3\zeta(4)}{8\pi^4} = \dfrac{1}{240} \approx 0.00364583333333333333333333333333333333333333333333333\ldots$

$\text{Li}_{-4}\left(\frac{1}{e}\right) - \Gamma(5) = \dfrac{e(e^3+11e^2+11e+1)}{(e-1)^5} \,-\, 24 \\ \\ \approx 0.0033329747690522721292121894483921207140518043451568930078000598\ldots$

$\dfrac{21\,\zeta(4)}{64\,\pi^4} = \dfrac{3\,\eta(4)}{8\,\pi^4} = \dfrac{7}{1920} \approx 0.003130145319788572754925768290785446702669\ldots$

$\zeta'(-9) = \dfrac{7129}{332640} \,-\, \ln A_9 \approx 0.003130145319788572754925768290785446702669\ldots$

$\dfrac{315\zeta(9)}{4\pi^9} \approx 0.002647116575123017994304712461106843854390356235263015431\ldots$

$\dfrac{80325\,\zeta(9)}{1024\,\pi^9} = \dfrac{315\,\eta(9)}{4\,\pi^9} \approx 0.002636776276001443705264459678055645245584143\ldots$

$\dfrac{3\zeta(5)}{4\pi^5} \approx 0.00254132611404785053204117740694788471052539657487680760322\ldots$

$\dfrac{45\,\zeta(5)}{64\,\pi^5} = \dfrac{3\,\eta(5)}{4\,\pi^5} \approx 0.00238249323191985987378860381901364191611755928894\ldots$

$\gamma_4 \approx 0.00232537006546730005746817017752606800090446941378485099075804\ldots$

$\dfrac{315\,\zeta(8)}{16\pi^8} = \dfrac{1}{480} \approx 0.002083333333333333333333333333333333333333333333333\ldots$

$\dfrac{40005\,\zeta(8)}{2048\,\pi^8} = \dfrac{315\,\eta(8)}{16\,\pi^8} = \dfrac{127}{61440} \approx 0.0020670572916666666666666666666666666\ldots$

$\gamma_3 \approx 0.00205383442030334586616004654275338428571580444541061824548148\ldots$

$\dfrac{15\zeta(6)}{8\pi^6} = \dfrac{1}{504} \approx 0.001984126984126984126984126984126984126984126984126\ldots$

$\dfrac{465\,\zeta(6)}{256\,\pi^6} = \dfrac{15\,\eta(6)}{8\,\pi^6} = \dfrac{31}{16128} \approx 0.001922123015873015873015873015873015873\ldots$

$\dfrac{45\zeta(7)}{8\pi^7} \approx 0.0018779516614843363839910052323044702985554170881081961291\ldots$

$\dfrac{2835\,\zeta(7)}{512\,\pi^7} = \dfrac{45\,\eta(7)}{8\,\pi^7} \approx 0.00184860866677364362799114577554971295014048869\ldots$

$\text{Li}_{-7}\left(\frac{1}{e}\right) - \Gamma(8) = \dfrac{e(e^6+120e^5+1191e^4+2416e^3+1191e^2+120e+1)}{(e-1)^8} \,- \nolinebreak 5040 \\ \\ \approx 0.00115211776836768480882153036833850684034239656299028269565500695\ldots$

$\gamma_1 - \dfrac{\ln 2}{2} - \dfrac{\ln 3}{6} + \dfrac{\ln^2 3}{2} \approx 0.00098299653128999412014625250027240285916350816\ldots$

$\gamma_5 \approx 0.00079332381730106270175333487744444483073153940458488707573425\ldots$

$\Gamma(-4)\;\zeta(-4) = \dfrac{\zeta(5)}{32\pi^4} \approx 0.0003326588104278593450290279499495543293849038\ldots$

$\Gamma(-7)\,\sin(4\pi) = \dfrac{\pi}{10080} \approx 0.0003116659378561302815935162086586808416862\ldots$

$\gamma_{10} \approx 0.0002053328149090647946837222892370653029598537741667643038402\ldots$

$\theta_3\left(0,\dfrac{1}{e}\right) \,-\, \sqrt{\pi} = 2\sqrt{\pi}\,e^{-\pi^2} \\ \\ \approx 0.00018335392113610036159077552692839413358953875053442385566778299\ldots$

$\dfrac{1}{2}\left(\theta_3\left(0,\dfrac{1}{e}\right)-\sqrt{\pi}\right) = \sqrt{\pi}\,e^{-\pi^2} \\ \\ \approx 0.00009167696056805018079538776346419706679476937526721192783389149 \ldots$

$\text{Li}_{-9}\left(\frac{1}{e}\right) - \Gamma(10) \\ \\ = \dfrac{e(e^8+502e^7+14608e^6+88234e^5+156190e^4+88234e^3+14608e^2+502e+1)}{(e-1)^{10}} \\ \\ - \, 362880 \approx 0.00004505588016251920838909027242634120738969477251877799\ldots$

$\Gamma(-8)\;\zeta(-8) = \dfrac{\zeta(9)}{512\pi^8} \approx 0.000000206254017500055738083443117820789043295\ldots$

$2\sqrt{\pi}\,e^{-(\frac{3\pi}{2})^2} \approx 0.0000000008042605927202409156553632043112141211750596603\ldots$

$\sqrt{\pi}\,e^{-(\frac{3\pi}{2})^2} \approx 0.00000000040213029636012045782768160215560706058752983018\ldots$

$\xi'(\frac{1}{2}) = \dfrac{\xi'(\frac{1}{2})}{\xi(\frac{1}{2})} = \sin(0) = \tan(0) = \cos\!\left(\dfrac{\pi}{2}\right)\! = \cot\!\left(\dfrac{\pi}{2}\right)\! = H_0 = 0$

$\Gamma(-6)\;\zeta(-6) = -\dfrac{\zeta(7)}{128\pi^6} \approx -0.000008194109921549913125874829733877813305\ldots$

$\gamma_9 \approx -0.0000343947744180880481779146237982273906207895385944416297592\ldots$

$\gamma_6 \approx -0.0002387693454301996098724218419080042777837151563580786314764\ldots$

$\gamma_8 \approx -0.0003521233538030395096020521650012087417291805337923503566573\ldots$

$\gamma_7 \approx -0.0005272895670577510460740975054788582819962534729698953310134\ldots$

$\zeta'(-5) = \dfrac{137}{15120} \,-\, \ln A_5 \approx -0.0005729859801986352049909941488338745132539\ldots$

$\dfrac{\zeta'(10)}{\zeta(10)} = \dfrac{93555}{\pi^{10}} \, \zeta'(10) = \ln(2\pi) + \gamma - 132 \ln A_9 \\ \\ \approx -0.000696340445284020436595276690937292736657676509203436446420355\ldots$

$\zeta'(10) = \zeta(10) \left(\ln(2\pi) + \gamma - 132 \ln A_9\right) = \dfrac{\pi^{10}}{93555} \left(\ln(2\pi) + \gamma - 132 \ln A_9\right) \\ \\ \approx -0.000697033008171393693680022578705010078649115457316074271577317\ldots$

$\zeta'(-7) = -\dfrac{121}{11200} \,-\, \ln A_7 \approx -0.00072864268015924065246723335465036061190\ldots$

$P'(9) \approx -0.0014104919214245312915541964563081999779016571316934961928\ldots$

$\text{Li}_{-5}\left(\frac{1}{e}\right) - \Gamma(6) = \dfrac{e(e^4+26e^3+66e^2+26e+1)}{(e-1)^6} \,-\, 120 \\ \\ \approx -0.002173232383975278537388051219268465151662206093269368904643546\ldots$

$P'(8) \approx -0.0028795247087292473913460284238573340649989837616758658410\ldots$

$\dfrac{\zeta'(8)}{\zeta(8)} = \dfrac{9450}{\pi^8} \, \zeta'(8) = \ln(2\pi) + \gamma + 240 \ln A_7 \\ \\ \approx -0.002890168308046756383549574847418599521209806619552036230054150\ldots$

$\text{Li}_{-6}\left(\frac{1}{e}\right) - \Gamma(7) = \dfrac{e(e^5+57e^4+302e^3+302e^2+57e+1)}{(e-1)^7} \,-\, 720 \\ \\ \approx -0.003064428353563511028785753367758666590548271949528132157866705\ldots$

$\zeta(-5) = -\dfrac{1}{252} \approx -0.00396825396825396825396825396825396825396825396825\ldots$

$\gamma_1 \,+\, \dfrac{\ln 2}{4}(2\ln 2 - 1) \approx -0.00587613366456233988134314307611297529133489612\ldots$

$\zeta'(-6) = -\dfrac{45\,\zeta(7)}{8\pi^6} = -\ln A_6 \approx -0.00589975914351593745062987740839202557\ldots$

$P'(7) \approx -0.0059406890391481961425505928290166090193681895059293510751\ldots$

$\zeta(-9) = -\dfrac{1}{132} \approx -0.00757575757575757575757575757575757575757575757575\ldots$

$\text{Li}_{-2}\left(\frac{1}{e}\right) - \Gamma(3) = \dfrac{e(e+1)}{(e-1)^3} \,-\, 2 \\ \\ \approx -0.007705232875012607073983386997882612168595476936229930476498315\ldots$

$\ln A_4 = -\dfrac{3}{4\pi^4} \, \zeta(5) \approx -0.00798381145026862428069667079878930390523769336\ldots$

$\ln A_8 = -\dfrac{315}{4\pi^8} \, \zeta(9) \approx -0.00831616198560224735952442651053421422567412291\ldots$

$\gamma_2 \approx -0.0096903631928723184845303860352125293590658061013407498807013\ldots$

$\xi'(0) = -\dfrac{1}{2} \,-\, \dfrac{\gamma}{4} \,+\, \dfrac{\ln(4\pi)}{4} \approx -0.01154785448306051690715512395324764581096\ldots$

$P'(6) \approx -0.0124590807227999915270277927746899700409113504715758758741\ldots$

$\dfrac{\zeta'(6)}{\zeta(6)} = \dfrac{945}{\pi^6} \, \zeta'(6) = \ln(2\pi) + \gamma - 252 \ln A_5 \\ \\ \approx -0.012633069032511060823892295945831999908383847500122255784102571\ldots$

$\zeta'(6) = \zeta(6) \left(\ln(2\pi) + \gamma - 252 \ln A_5\right) = \dfrac{\pi^6}{945} \left(\ln(2\pi) + \gamma - 252 \ln A_5\right) \\ \\ \approx -0.012852165131795725075945401459923608828848753911931951367969449\ldots$

$\Gamma(-5)\,\sin(3\pi) = -\dfrac{\pi}{240} \approx -0.0130899693899574718269276807636645953508215\ldots$

$\Gamma(-2)\;\zeta(-2) = -\dfrac{\zeta(3)}{8\pi^2} \approx -0.0152242285291966353901257652355773883235002\ldots$

$\text{Ei}\!\left(\dfrac{1}{e}\right) \approx -0.01810250634715775421599279350392527458302421140777456216\ldots$

$-\dfrac{1}{2} \, \Psi^{(0)}\!\left(\dfrac{3}{2}\right) = \dfrac{\gamma}{2} \,-\, \dfrac{H_\frac{1}{2}}{2} \approx -0.018244986989288260279511833500622216403420\ldots$

$\zeta'(-10) = -\dfrac{14175\,\zeta(11)}{8\pi^{10}} = -\ln A_{10} \approx -0.01892992633814037422898050229034\ldots$

$\dfrac{\xi'(10)}{\xi(10)} + \dfrac{\zeta'(-9)}{\zeta(-9)} = \dfrac{1}{2} H_5 - H_8 + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} = -\dfrac{331}{210} + \dfrac{\ln(4\pi)}{2} + \dfrac{\gamma}{2} \\ \\ \approx -0.022070520255064363683988634014569051055963267402603351195815313\ldots$

$\dfrac{\xi'(0)}{\xi(0)} = -\dfrac{\xi'(1)}{\xi(1)} = 2\,\xi'(0) = -2\,\xi'(1) = -1 + \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx -0.023095708966121033814310247906495291621932127152050759525392072\ldots$

$-\dfrac{15\;\zeta(5)}{2\pi^{5}} \approx -0.0254132611404785053204117740694788471052539657487680760\ldots$

$\zeta(-\frac{3}{2}) = -\dfrac{3}{16\pi^2} \, \zeta(\frac{5}{2}) \approx -0.025485201889833035949542986910704745469024984\ldots$

$P'(5) \approx -0.0268386012767983574221875132924501599433301495535582281248\ldots$

$\dfrac{\ln\pi}{2}\,-\,3\,-\,\dfrac{\ln 2}{2}\,+\,\dfrac{5\ln 3}{2} \approx -0.02767792568499833914878929274624466659537626\ldots$

$\dfrac{1}{\sqrt{2}} \left(\zeta(-\frac{1}{2})+\dfrac{1}{6}\right) \approx -0.029146629199001724965238160136406242985162102221\ldots$

$\zeta'(-2) = -\dfrac{\zeta(3)}{4\pi^2} = -\ln A_2 \approx -0.0304484570583932707802515304711547766470\ldots$

$H_{-\frac{\pi}{2}} \approx -0.03259686937614189516312738735001334130268817312652775679552\ldots$

$\dfrac{\zeta'(7)}{\zeta(7)} + \dfrac{\zeta'(-6)}{\zeta(-6)} = -H_6 + \ln(2\pi) + \gamma = -\dfrac{49}{20} + \ln(2\pi) + \gamma \\ \\ \approx -0.034907268689121655832828437106362289235045716784509575559929684\ldots$

$\xi'(-1) = -\,\xi'(2) = \dfrac{\pi}{6} \left(-\dfrac{3}{2} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} + 12\ln{A}\right) \\ \\ \approx -0.036162994264296978312750475609936708999820602923277231234985271\ldots$

$-\dfrac{1+\zeta'(0)}{2} = \dfrac{\ln(2\pi)}{4}\,-\,\dfrac{1}{2} \approx -0.04053073339766362910983513179719118006930\ldots$

$\zeta(-\frac{1}{2}) \,+\, \dfrac{1}{6} \approx -0.041219558310687899350640058730382635559601864621005870\ldots$

$\dfrac{\ln\pi}{2}\,+\,2\ln 2\,-\,2 \approx -0.04134069595540929409382208140711750802535232482183\ldots$

$-\dfrac{5}{4} \,-\, \dfrac{\gamma}{2} \,+\, 6\ln{A} \approx -0.0460809702480608550197380835845173589368613859587\ldots$

$1\,-\, \dfrac{\pi^2}{8}\,+\,\gamma^2\,+\,2\gamma_1 \approx -0.0461543172958046027571079903790773035302679623241\ldots$

$-\dfrac{2835\;\zeta(9)}{2\pi^{9}} \approx -0.04764809835221432389748482429992318937902641223473427\ldots$

$\ln\!\left(\dfrac{8}{3}\sqrt{\dfrac{2}{5\pi}}\right) = -\dfrac{1}{2}\ln\!\left(\dfrac{\pi}{2}\right) + 3\ln 2 - \ln 3 - \dfrac{\ln 5}{2} \\ \approx -0.049681055850051382807026154108530891969688171153770709025449781\ldots$

$\dfrac{\zeta'(-2)}{\xi(-2)} = -\dfrac{1}{6\pi} \approx -0.05305164769729844525629458779083812067815321524681\ldots$

$\dfrac{1}{2\pi}\,\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{2\pi}\right) = \dfrac{H_\frac{1}{2\pi} - \gamma}{2\pi} \\ \\ \approx -0.054443243388232951061075924125608110158078683738986494412102405\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{\ln 2}{2} \approx -0.05796575782920622440536001568788706851667039921016582765\ldots$

$\Gamma(-\frac{9}{2}) = -\dfrac{32\sqrt{\pi}}{945} \approx -0.060019601300504246427027893615784810422774161477\ldots$

$P'(4) \approx -0.0606076333507700633922309837097133784063828774612598439911\ldots$

$\dfrac{1}{2} \ln\!\left(\dfrac{\pi}{2}\right) \,-\, 2\ln 2 \,+\, \ln 3 \approx -0.0618907198070534950761213910463863597176123\ldots$

$\dfrac{\zeta'(4)}{\zeta(4)} = \dfrac{90}{\pi^4} \, \zeta'(4) = \ln(2\pi) + \gamma + 120 \ln A_3 \\ \\ \approx -0.063669764955371126496198675689356770048111819193303756026905804\ldots$

$\zeta'(4) = \zeta(4) \left(\ln(2\pi) + \gamma + 120 \ln A_3\right) = \dfrac{\pi^4}{90} \left(\ln(2\pi) + \gamma + 120 \ln A_3\right) \\ \\ \approx -0.068911265896125379848829365587440827150016374871378463827585706\ldots$

$\dfrac{\xi'(-1)}{\xi(-1)} = -\dfrac{\xi'(2)}{\xi(2)} = -\dfrac{3}{2} - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} + 12\ln{A} \\ \\ \approx -0.069066231530000676225165919262539426251790644765451435282107486\ldots$

$\gamma_1 = \gamma_1(1) = \gamma_1(2) \approx -0.0728158454836767248605863758749013191377363383\ldots$

$\dfrac{1}{2}\,-\,\gamma \approx -0.0772156649015328606065120900824024310421593359399235988057\ldots$

$-\dfrac{1}{2}\,+\,\dfrac{\ln(2\pi)}{2}\,-\,2\ln A \approx -0.078570420862895783314176250746610312333341953\ldots$

$\text{Li}_{-1}\left(\frac{1}{e}\right) - \Gamma(2) = \dfrac{e}{(e\!-\!1)^2} \,-\, 1 \\ \\ \approx -0.079326405792207681054586477283500397118344373349448847646039590\ldots$

$-\zeta'(0) - 1 = \dfrac{\ln(2\pi)}{2} \,-\, 1 \approx -0.0810614667953272582196702635943823601386025\ldots$

$\dfrac{1}{2}\,\ln\!\left(\dfrac{8}{3\pi}\right) = \dfrac{1}{2}\left(3\ln 2 - \ln\pi - \ln 3\right) \approx -0.0819503164188369686434881119505\ldots$

$\zeta(-1) = -\dfrac{1}{12} \approx -0.083333333333333333333333333333333333333333333333333\ldots$

$\dfrac{1}{2} \cot\dfrac{1}{2} \,-\, 1 \approx -0.084756139143774040365990280515591688120946025991932997\ldots$

$\dfrac{1}{3}\,-\,\dfrac{\ln(2\pi)}{2}\,+\,2\ln A \approx -0.08809624580377088335249041592005635433332471296\ldots$

$M_2 \,-\, \gamma \,+\, \ln\gamma \approx -0.092101095985739771324379560420671964702550332858603\ldots$

$\dfrac{1\!-\!\zeta(3)}{2} \approx -0.10102845157979714269986908075572499538249314617024944089\ldots$

$2\ln 2 \,-\, \dfrac{3}{2} \approx -0.1137056388801093811655357570836468638489997312794894917\ldots$

$\gamma \,-\, \ln 2 \approx -0.11593151565841244881072003137577413703334079842033165531\ldots$

$\dfrac{\gamma}{2} \,+\, \ln 2 \,-\, \ln 3 \approx -0.116857275657397951674757070423147921050910755492532\ldots$

$\ln\!\left(\dfrac{\sqrt{\pi}}{2}\right) = \dfrac{\ln\pi}{2} \,-\, \ln 2 \approx -0.1207822376352452223455184457816472122518527\ldots$

$\dfrac{1}{\sqrt{2}} \, \zeta(-\frac{1}{2}) \approx -0.146997759396759645698712220487214416199301425169869947\ldots$

$P'(3) \approx -0.1507575555439504221798365163653429195755011615306893318187\ldots$

$-\dfrac{\ln 2}{2} - \dfrac{\ln 3}{2} - \ln(2\pi) + 2\ln\Gamma\!\left(\dfrac{1}{4}\right) \\ \\ \approx -0.157711751627218069225677271562151824233535163141348077562016207\ldots$

$\dfrac{\zeta(-1)}{\xi(-1)} = -\dfrac{1}{2\pi} \approx -0.159154943091895335768883763372514362034459645740456\ldots$

$\ln\!\left(\dfrac{8}{3\pi}\right) = 3\ln 2 \,-\, \ln\pi \,-\, \ln 3 \approx -0.16390063283767393728697622390105471206\ldots$

$\dfrac{\zeta'(3)}{\zeta(3)} \approx -0.16482268215827724018649339479947506502300958042240327087174\ldots$

$\zeta'(-1) = \dfrac{1}{12} \,-\, \ln{A} \approx -0.1654211437004509292139196602427806427640363803\ldots$

$\dfrac{\gamma}{2} \,+\, \dfrac{\ln 2}{2} \,-\, \dfrac{\ln 3}{2} \approx -0.16953753348631110228850756084280432020397094198416\ldots$

$\dfrac{1}{4}\,-\,\dfrac{\ln(2\pi)}{2}\,+\,2\ln A \approx -0.1714295791371042166858237492533896876666580463\ldots$

$\ln(\sin1) \approx -0.17260374626909167851341097586390906984010840889640480490\ldots$

$\dfrac{\zeta'(-7)}{\zeta(-7)} = -\dfrac{363}{140} - 240\,\ln A_7 \\ \\ \approx -0.174874243238217756592136005116086546856693053022100396472732676\ldots$

$\dfrac{\zeta'(8)}{\zeta(8)} + \dfrac{\zeta'(-7)}{\zeta(-7)} = -H_7 + \ln(2\pi) + \gamma = -\dfrac{363}{140} + \ln(2\pi) + \gamma \\ \\ \approx -0.177764411546264512975685579963505146377902859641652432702786827\ldots$

$\dfrac{1}{2} \,-\, \ln 2 \approx -0.19314718055994530941723212145817656807550013436025525412\ldots$

$\zeta'(3) \approx -0.19812624288563685333068182150328579687554279346383500334689\ldots$

$\zeta(-\frac{1}{2}) = -\dfrac{\zeta(\frac{3}{2})}{4\pi} \approx -0.2078862249773545660173067253970493022262685312876\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{1}}{2} = \dfrac{\gamma-1}{2} \approx -0.2113921675492335696967439549587987844789203320\ldots$

$\text{Ei}(-1) = -E_1(1) = -\dfrac{G}{e} \approx -0.21938393439552027367716377546012164903104\ldots$

$-\dfrac{1}{2} \ln\!\left(\dfrac{\pi}{2}\right) = \dfrac{1}{2} \ln\!\left(\dfrac{2}{\pi}\right) \approx -0.2257913526447274323630976149474410717858973\ldots$

$-\dfrac{\ln^2 2}{2} \approx -0.240226506959100712333551263163332485865276475797272793433\ldots$

$\ln\!\left(\dfrac{\pi}{4}\right) = \ln\pi \,-\, 2\ln 2 \approx -0.241564475270490444691036891563294424503705455\ldots$

$2\ln\pi \,-\, 10\ln 2 \,+\, 4\ln 3 \approx -0.24756287922821398030448556418554543887044948\ldots$

$-\dfrac{2}{\pi^2} (H_\frac{\pi}{2} + H_{-\frac{\pi}{2}}) \approx -0.2597696710475819131824844816281819875803001855518\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{\ln 3}{2} \approx -0.26069831188328841539436657342006163680266561094141292646\ldots$

$\zeta(-\frac{1}{3}) \approx -0.2773430478401295269760914767751689778438482439659731975854\ldots$

$\ln\zeta(0) = -\dfrac{\gamma}{2} \approx -0.288607832450766430303256045041201215521079667969961\ldots$

$3(1-\ln3) \approx -0.2958368660043290741857357107675771139424716734682483552\ldots$

$\dfrac{\zeta'(9)}{\zeta(9)} + \dfrac{\zeta'(-8)}{\zeta(-8)} = -H_8 + \ln(2\pi) + \gamma = -\dfrac{761}{280} + \ln(2\pi) + \gamma \\ \\ \approx -0.302764411546264512975685579963505146377902859641652432702786827\ldots$

$\dfrac{\zeta(0)}{\zeta(2)} = -\dfrac{1}{2\pi^2} \, \dfrac{B_0}{B_2} = -\dfrac{3}{\pi^2} \approx -0.303963550927013314331638389629182916713072\ldots$

$-\dfrac{\ln(2\pi)}{6} \approx -0.30631284440155758059344324546853921328713249121259447093\ldots$

$\ln 2 \,-\, 1 \approx -0.30685281944005469058276787854182343192449986563974474587\ldots$

$H_{-\frac{1}{6}} = \dfrac{\pi\sqrt{3}}{2} \,-\, 2\ln 2 \,-\, \dfrac{3\ln 3}{2} \approx -0.313513747770728380036214711836908094696\ldots$

$\dfrac{4\ln 2}{3} + \dfrac{3\ln\pi}{2} - 3\ln\Gamma\!\left(\dfrac{1}{3}\right) \\ \\ \approx -0.314970871262607200456738255293394016968393757304349866606263077\ldots$

$M_1 \,-\, \gamma \approx -0.315718452053890076851085251473706571990592687678724392613\ldots$

$\zeta(-\frac{1}{4}) \approx -0.3204512642285772827904444930548724697059108390851109327114\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{6} = \ln\!\dfrac{\sqrt{\pi}}{\sqrt{6}} \approx -0.32351479168932741333452500351382178053784793963384\ldots$

$\gamma\,-\,\dfrac{\ln(2\pi)}{2} \approx -0.341722868303139881173817646323215208819238137697859814\ldots$

$\Psi^{(0)}\!\left(1\!+\!\dfrac{1}{2\pi}\right) = H_\frac{1}{2\pi} - \gamma \approx -0.342076986932147445072680974261289823013497\ldots$

$\dfrac{\zeta'(3)}{\xi(3)} = \dfrac{2\pi}{3} \, \dfrac{\zeta'(3)}{\zeta(3)} \approx -0.3452038182756061762594888561608416497423424070194\ldots$

$\zeta(-\frac{1}{5}) \approx -0.3496662805983141371352638156539947269601967570331519754996\ldots$

$-\dfrac{1}{2} \, \Psi^{(0)}\!\left(\dfrac{5}{2}\right) = \dfrac{\gamma}{2} \,-\, \dfrac{H_\frac{3}{2}}{2} \approx -0.351578320322621593612845166833955549736753\ldots$

$\zeta'(-\frac{1}{2}) \approx -0.360854339599947607347420806363951065884852787918632210814\ldots$

$\zeta(-\frac{1}{6}) \approx -0.3707376572047620140371679962990478051365904480790641335601\ldots$

$1\,-\,2\ln 2 \approx -0.3862943611198906188344642429163531361510002687205105082\ldots$

$\zeta(-\frac{1}{7}) \approx -0.3866427317653044093652249045766036660082834933025099293987\ldots$

$H_{-\frac{1}{5}} = H_\frac{1}{5} - 5 + \pi\cot(\frac{\pi}{5}) = \dfrac{\pi}{2} \sqrt{1+\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} - \dfrac{\sqrt{5}}{4} \ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx -0.387792901804605598784785543074432988592001153755299230304043559\ldots$

$\ln\!\left(\dfrac{2\pi\sqrt{2}}{\Gamma^2(\frac{1}{4})}\right) = 2\ln\!\left(\dfrac{2^\frac{1}{4}\sqrt{2\pi}}{\Gamma(\frac{1}{4})}\right) = \ln(2\pi) + \dfrac{\ln 2}{2} - 2\ln\Gamma(\frac{1}{4}) \\ \\ \approx -0.391594392706836776471945346899111028090210115770026648305330959\ldots$

$\zeta(-\frac{1}{8}) \approx -0.3990696689450450355098692830142123540028063746889528856365\ldots$

$\dfrac{\gamma}{2} \,-\, \ln 2 \approx -0.40453934810917887911397607641697535255442046639029345471\ldots$

$\zeta(-\frac{1}{9}) \approx -0.4090449715146988251599454388039843390392539452226351638498\ldots$

$\dfrac{\zeta'(-9)}{\zeta(-9)} = -\dfrac{7129}{2520} + 132\,\ln A_9 \\ \\ \approx -0.413179182212091603650201414383678964752356294243560107367477583\ldots$

$\dfrac{\zeta'(10)}{\zeta(10)} + \dfrac{\zeta'(-9)}{\zeta(-9)} = -H_9 + \ln(2\pi) + \gamma = -\dfrac{7129}{2520} + \ln(2\pi) + \gamma \\ \\ \approx -0.413875522657375624086796691074616257489013970752763543813897938\ldots$

$\zeta(-\frac{1}{10}) \approx -0.417228040767366856808361420078239751280306985555838610354\ldots$

$\text{Li}_0\left(\frac{1}{e}\right) - \Gamma(1) = \dfrac{1}{e-1} \,-\, 1 \\ \\ \approx -0.418023293130673575614997994890988441453130698924603863733212940\ldots$

$\dfrac{1-\ln(2\pi)}{2} \approx -0.418938533204672741780329736405617639861397473637783412\ldots$

$\gamma_1(3) = \gamma_1 - \dfrac{\ln 2}{2} \approx -0.4193894357636493795692024366039896031754864055144\ldots$

$-\Psi^{(0)}(2) = \zeta(1) \,-\, H_1 = \gamma \,-\, 1 \approx -0.4227843350984671393934879099175975689\ldots$

$-\ln\!\left(\dfrac{\pi}{2}\right) = \ln\!\left(\dfrac{2}{\pi}\right) \approx -0.4515827052894548647261952298948821435717946785\ldots$

$H_{-\frac{3\pi}{2}} \approx -0.4572844791248549775282738028956225525648006347678181558173\ldots$

$1 \,-\, \ln(e-1) \approx -0.45867514538708189102164364506732970187697790669219414\ldots$

$\cos1 \,-\, 1 \approx -0.4596976941318602825990633925570233962676895793820777723\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{2}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{3}{4} \approx -0.4613921675492335696967439549587987844789203320\ldots$

$\dfrac{1}{2} \ln\!\left(\dfrac{\pi}{8}\right) = \dfrac{1}{2} \ln\!\left(\dfrac{\pi}{2}\right) \,-\, \ln 2 \approx -0.4673558279152178770541345065107354962896\ldots$

$\dfrac{1}{2}\,-\,\ln 2\,-\,\dfrac{\gamma}{2} \approx -0.4817550130107117397204881664993777835965798023302170\ldots$

$\ln\!\left(\dfrac{\pi^2}{16}\right) = 2\ln\pi \,-\, 4\ln 2 \approx -0.483128950540980889382073783126588849007410\ldots$

$P'(2) \approx -0.4930911093687644621978262050564912580555881263464682907133\ldots$

$\zeta(0) = -\eta(0) = -\xi(0) = -\xi(1) = P(0) = -\dfrac{1}{4} \csc^2\!\left(\dfrac{\pi}{4}\right) = -0.5$

$H_{-\frac{1}{4}} = H_\frac{1}{4} - 4 + \pi\cot(\frac{\pi}{4}) = \dfrac{\pi}{2} - 3\ln 2 \\ \\ \approx -0.508645214884939309020374672734778262127915703393212851874567732\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{\ln 5}{2} \approx -0.51611112376628375699712362157189260424172100916429706155\ldots$

$\dfrac{11\ln 2}{6} - \dfrac{\ln 3}{2} + \dfrac{3\ln\pi}{2} - 3\ln\Gamma\!\left(\dfrac{1}{3}\right) \\ \\ \approx -0.517703425316689391445744813025568585254388969035596965413270239\ldots$

$\gamma \,-\, \ln 3 \approx -0.52139662376657683078873314684012327360533122188282585292\ldots$

$-\dfrac{6081075\;\zeta(13)}{4\pi^{13}} \approx -0.5236309567372813746331887469610731999363879908155\ldots$

$-\dfrac{\sqrt{\pi}}{2} \, \text{erfi}\left(\dfrac{1}{2}\right) \approx -0.54498710418362222366242013140701667104079810362893\ldots$

$\ln\zeta(1) = \ln\gamma \approx -0.549539312981644822337661768802907788330698981263064\ldots$

$-\dfrac{1}{2} \, \Psi^{(0)}\!\left(\dfrac{7}{2}\right) = \dfrac{\gamma}{2} \,-\, \dfrac{H_\frac{5}{2}}{2} \approx -0.551578320322621593612845166833955549736753\ldots$

$\dfrac{\xi'(1)}{\xi(1)} - \dfrac{\zeta'(1)}{\zeta(1)} = \dfrac{1}{2} H_\frac{1}{2} - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = 1 - \dfrac{\ln(4\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx -0.554119955935411826792201842175907139420227208787872839280375162\ldots$

$1-\tan 1 \approx -0.55740772465490223050697480745836017308725077238152003838\ldots$

$\dfrac{\zeta'(2)}{\zeta(2)} = \ln(2\pi) + \gamma - 12\ln{A} \\ \\ \approx -0.569960993094532806399864360019730002403482280806930979558197360\ldots$

$\Psi^{(0)}(1) = \dfrac{\Gamma'(1)}{\Gamma(1)} = \Gamma'(1) = H_0 - \gamma = \Psi^{(0)}(0) = \dfrac{\Gamma'(0)}{\Gamma(0)} = H_{-1} - \gamma \\ \\ = -\gamma \approx -0.5772156649015328606065120900824024310421593359399235988057\ldots$

$\zeta(\frac{1}{10}) \approx -0.6030375198562417152484319382634382079141478245521286336667\ldots$

$-\dfrac{\ln^2 3}{2} \approx -0.603474480406290988921889561924682956809231673314610992408\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{\ln 6}{2} \approx -0.6072719021632610701029826341491499208404156781215405535\ldots$

$\ln(\cos1) \approx -0.6156264703860142621470375164088918633509354239463728341\ldots$

$\zeta(\frac{1}{9}) \approx -0.6160334664611776165475574281522051848167437277671587784557\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{3}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{11}{12} \approx -0.62805883421590023636341062162546545114558699\ldots$

$\zeta(\frac{1}{8}) \approx -0.6327756234986952552935252676356462715268637913112166822606\ldots$

$1 \,-\, \dfrac{\pi^2}{6} \approx -0.64493406684822643647241516664602518921894990120679843773\ldots$

$\zeta(\frac{1}{7}) \approx -0.6551535579715974093047713865534759227571676145703814978496\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{\ln 7}{2} \approx -0.684347242076890222249420326680388649297462696820968794\ldots$

$\zeta(\frac{1}{6}) \approx -0.6865815819473582278366765812466125231345505207235060904798\ldots$

$-\ln 2 \approx -0.6931471805599453094172321214581765680755001343602552541206\ldots$

$-\dfrac{1}{2} \, \Psi^{(0)}\!\left(\dfrac{9}{2}\right) = \dfrac{\gamma}{2} \,-\, \dfrac{H_\frac{7}{2}}{2} \approx -0.694435463179764450755702309691098406879610\ldots$

$\dfrac{1}{e} \left(\text{Ei}(\frac{1}{e}) - \text{Ei}(1)\right) \approx -0.70383442315386095104012793401897255742733337311\ldots$

$\zeta(\frac{1}{5}) \approx -0.7339209248963405922438096137551368666491286938323153364439\ldots$

$H_{-\frac{1}{3}} = H_\frac{1}{3} - 3 + \pi\cot(\frac{\pi}{3}) = \dfrac{\pi}{2\sqrt{3}} - \dfrac{3\ln 3}{2} \\ \\ \approx -0.741018750885055611795828726562710690829202712687753889817099032\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{3\ln 2}{2} \approx -0.751112938389151533822592137146063636592170533570421081\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{4}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{25}{24} \approx -0.75305883421590023636341062162546545114558699\ldots$

$\gamma_1(4) = \gamma_1 - \dfrac{\ln 2}{2} - \dfrac{\ln 3}{3} \approx -0.78559353198635261003428418224483150472464992\ldots$

$-\dfrac{1}{2} \, \Psi^{(0)}\!\left(\dfrac{11}{2}\right) = \dfrac{\gamma}{2} \,-\, \dfrac{H_\frac{9}{2}}{2} \approx -0.80554657429087556186681342080220951799072\ldots$

$\gamma \,-\, 2\ln 2 \approx -0.809078696218357758227952152833950705108840932780586909\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{2\ln 3}{2} \approx -0.810004456217343261091989191881324489126410889852787652\ldots$

$\zeta(\frac{1}{4}) \approx -0.8132784052618916565214478200735255744815705245290058426050\ldots$

$\text{Li}_2(-1) = -\eta(2) = -\dfrac{\pi^2}{12} \approx -0.82246703342411321823620758332301259460947\ldots$

$-\sin1 \approx -0.841470984807896506652502321630298999622563060798371065672\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{5}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{137}{120} \approx -0.85305883421590023636341062162546545114558699\ldots$

$\dfrac{\gamma}{2} \,-\, \dfrac{\ln{10}}{2} \approx -0.8626847140462564117057396823009808882794710763444246886\ldots$

$\dfrac{4\ln 2}{3} - \dfrac{\ln 3}{2} + \dfrac{3\ln\pi}{2} - 3\ln\Gamma\!\left(\dfrac{1}{3}\right) \\ \\ \approx -0.864277015596662046154360873754656869292139036215724592473610243\ldots$

$P_1 = M_1 \,-\, \gamma \,+\, \ln\gamma \approx -0.8652577650355348991887470202766143603212916689\ldots$

$\text{arsinh}(-1) = -\text{arsinh}(1) = -\ln(1+\sqrt{2}) \\ \\ \approx -0.881373587019543025232609324979792309028160328261635410753295608\ldots$

$2\zeta(3) \,-\, \dfrac{\pi^2}{3} \approx -0.885754327377264302145354010269150396907927217732599111\ldots$

$\dfrac{1}{2}\,-\,2\ln 2 \approx -0.8862943611198906188344642429163531361510002687205105082\ldots$

$-2\ln\!\left(\dfrac{\pi}{2}\right) = 2\ln\!\left(\dfrac{2}{\pi}\right) \approx -0.90316541057890972945239045978976428714358935\ldots$

$\zeta'(0) = -\dfrac{\ln(2\pi)}{2} \approx -0.918938533204672741780329736405617639861397473637\ldots$

$\zeta(1) \,-\, H_2 = \gamma \,-\, \dfrac{3}{2} \approx -0.9227843350984671393934879099175975689578406640\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{6}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{49}{40} \approx -0.93639216754923356969674395495879878447892033\ldots$

$\zeta'(2) = \zeta(2) \, (\gamma + \ln{(2\pi)} - 12\ln{A}) = \dfrac{\pi^2}{6} \, (\gamma + \ln{(2\pi)} - 12\ln{A}) \\ \\ \approx -0.93754825431584375370257409456786497789786028861482992588543348\ldots$

$\Gamma(-\frac{5}{2}) = -\dfrac{8\sqrt{\pi}}{15} \approx -0.945308720482941881225689324448610764158693043265\ldots$

$\dfrac{1\,-\,\sqrt{\pi}\,\text{erfi}(1)}{2} \approx -0.9626517459071816088040485868569881551208700962167\ldots$

$H_{-\frac{2}{5}} = H_\frac{2}{5} - \dfrac{5}{2} + \pi\cot(\frac{2\pi}{5}) = \dfrac{\pi}{2} \sqrt{1-\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} + \dfrac{\sqrt{5}}{4} \ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx -0.96340354899165755415415185872382951046837491874968360946091802\ldots$

$\zeta(\frac{1}{3}) \approx -0.9733602483507827154688868624478965707728296317430533399453\ldots$

$\dfrac{\xi(0)}{\zeta(0)} = \dfrac{\zeta(0)}{\xi(0)} = -1$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{7}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{363}{280} \approx -1.0078207389778049982681725263873702130503489\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{24} = \ln\!\dfrac{\sqrt{\pi}}{2\sqrt{6}} \approx -1.016661972249272722751757124971998348613348073994\ldots$

$\gamma \,-\, \ln 5 \approx -1.0322222475325675139942472431437852084834420183285941231\ldots$

$\ln\dfrac{6\gamma}{\pi^2} \approx -1.047239615452390169812039113128322938902297914910683228262\ldots$

$\dfrac{11\ln 2}{6} - \ln 3 + \dfrac{3\ln\pi}{2} - 3\ln\Gamma\!\left(\dfrac{1}{3}\right) \\ \\ \approx -1.067009569650744237143367431486831437578134247946971691280617405\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{8}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{761}{560} \approx -1.0703207389778049982681725263873702130503489\ldots$

$-\dfrac{1}{2} \,-\, \gamma \approx -1.07721566490153286060651209008240243104215933593992359880\ldots$

$-2\left(\dfrac{1}{3}\,-\,\zeta(-\frac{1}{2})\right) \approx -1.0824391166213757987012801174607652711192037292\ldots$

$\Psi^{(0)}(\frac{3}{4}) = \dfrac{\Gamma'(\frac{3}{4})}{\Gamma(\frac{3}{4})} = H_{-\frac{1}{4}} - \gamma = \dfrac{\pi}{2} - 3\ln 2 - \gamma \\ \\ \approx -1.08586087978647216962688676281718069317007503933313645068033496\ldots$

$-\dfrac{1}{4}\,\csc^2\!\left(\dfrac{1}{2}\right) \approx -1.087671324835010705388393379828034987924223733812092\ldots$

$-\sqrt{\pi} \, \text{erfi}\!\left(\dfrac{1}{2}\right) \approx -1.0899742083672444473248402628140333420815962072578\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{9}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{7129}{5040} \approx -1.125876294533360553823728081942925768605904\ldots$

$\dfrac{\zeta'(2)}{\zeta(2)} - \dfrac{\zeta'(1)}{\zeta(1)} = \ln(2\pi) - 12 \ln A \\ \\ \approx -1.14717665799606566700637645010213243344564161674685457836396459\ldots$

$\dfrac{\zeta(1)}{2} \,-\, \dfrac{H_{10}}{2} = \dfrac{\gamma}{2} \,-\, \dfrac{7381}{5040} \approx -1.175876294533360553823728081942925768605904\ldots$

$-\dfrac{1}{2}\,-\,\ln 2 \approx -1.193147180559945309417232121458176568075500134360255254\ldots$

$\gamma \,-\, \ln 6 \approx -1.2145438043265221402059652682982998416808313562430811070\ldots$

$\ln\!\left(\tan\left(\dfrac{\pi}{4}-\dfrac{1}{2}\right)\!\right) = -\ln\!\left(\tan\left(\dfrac{\pi}{4}+\dfrac{1}{2}\right)\!\right) = \ln(\sec1 - \tan1) \\ \\ \approx -1.22619117088351707081306096747190675272424835022074027913861684\ldots$

$\zeta(1) \,-\, H_3 = \gamma \,-\, \dfrac{11}{6} \approx -1.256117668431800472726821243250930902291173997\ldots$

$\dfrac{\zeta'(1)}{\zeta(1)} - \dfrac{\zeta'(0)}{\zeta(0)} = \gamma - \ln(2\pi) \\ \\ \approx -1.26066140150781262295414738272883284868063561133564322682853584\ldots$

$-M_3 \approx -1.332582275733220881765828776071027748838459489042422661787\ldots$

$\gamma \,-\, \ln 7 \approx -1.3686944841537804444988406533607772985949253936419375896\ldots$

$H_{-\frac{1}{2}} = H_\frac{1}{2} - 2 + \pi\cot(\frac{\pi}{2}) = -2\ln 2 \\ \\ \approx -1.386294361119890618834464242916353136151000268720510508241360018\ldots$

$\gamma\,-\,2 \approx -1.422784335098467139393487909917597568957840664060076401194\ldots$

$\zeta(\frac{1}{2}) \approx -1.4603545088095868128894991525152980124672293310125814905428\ldots$

$-\dfrac{\sqrt{\pi}}{2} \, \text{erfi}(1) \approx -1.462651745907181608804048586856988155120870096216739\ldots$

$\gamma \,-\, 3\ln 2 \approx -1.502225876778303067645184274292127273184341067140842163\ldots$

$\dfrac{\zeta'(\frac{3}{2})}{\zeta(\frac{3}{2})} \approx -1.5052353557882679194220436030392749710400434432200244582615\ldots$

$\zeta(1) \,-\, H_4 = \gamma \,-\, \dfrac{25}{12} \approx -1.506117668431800472726821243250930902291173997\ldots$

$4(1-\ln 4) \approx -1.5451774444795624753378569716654125446040010748820420329\ldots$

$\Gamma(-1)\;\sin\pi = -\dfrac{\pi}{2} \approx -1.5707963267948966192313216916397514420985846996\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{80} = \ln\!\dfrac{\sqrt{\pi}}{4\sqrt{5}} \approx -1.618648374412240719063130233852917600090153539397\ldots$

$\gamma \,-\, 2\ln 3 \approx -1.620008912434686522183978383762648978252821779705575304\ldots$

$-\dfrac{\zeta(4)}{\xi(4)} = -\text{Li}_2(1) = -\zeta(2) = -\dfrac{\pi^2}{6} \\ \approx -1.644934066848226436472415166646025189218949901206798437735558229\ldots$

$\text{erfi}(-1) = -\text{erfi}(1) \approx -1.6504257587975428760253377295613624438956798748\ldots$

$-\dfrac{\zeta'(-1)}{\zeta(-1)} = -1 - 2\ln(2\pi) + 12\ln{A} \\ \\ \approx -1.690700408413279816554283022709102846277153330528712247270338485\ldots$

$-1\,-\,\ln 2 \approx -1.6931471805599453094172321214581765680755001343602552541\ldots$

$\zeta(1) \,-\, H_5 = \gamma \,-\, \dfrac{137}{60} \approx -1.706117668431800472726821243250930902291173997\ldots$

$\gamma \,-\, \ln{10} \approx -1.7253694280925128234114793646019617765589421526888493772\ldots$

$2\,\zeta'(0) = -\ln(2\pi) \approx -1.83787706640934548356065947281123527972279494727\ldots$

$-\ln 2\,-\,2\gamma \approx -1.84757851036301103063025630162298143015981880624010245\ldots$

$\dfrac{\xi'(0)}{\xi(0)} - \dfrac{\zeta'(0)}{\zeta(0)} = \dfrac{1}{2} H_0 - 1 - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} = -1 - \dfrac{\ln\pi}{2} - \dfrac{\gamma}{2} \\ \\ \approx -1.860972775375466517374969720717730571344727074427617585159695153\ldots$

$\zeta(1) \,-\, H_6 = \gamma \,-\, \dfrac{49}{20} \approx -1.872784335098467139393487909917597568957840664\ldots$

$4 \,-\, \dfrac{17\ln2}{2} \approx -1.89175103475953513004647303239450082864175114206216966\ldots$

$1 \,-\, \sqrt{\pi}\,\text{erfi}(1) \approx -1.925303491814363217608097173713976310241740192433478\ldots$

$\Psi^{(0)}(\frac{1}{2}) = \dfrac{\Gamma'(\frac{1}{2})}{\Gamma(\frac{1}{2})} = H_{-\frac{1}{2}} - \gamma = -2\ln 2 - \gamma \\ \\ \approx -1.96351002602142347944097633299875556719315960466043410704712725\ldots$

$H_{-\frac{3}{5}} = H_\frac{3}{5} - \dfrac{5}{3} + \pi\cot(\frac{3\pi}{5}) = -\dfrac{\pi}{2} \, \sqrt{1-\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} + \dfrac{\sqrt{5}}{4} \, \ln \dfrac{3\!+\!\sqrt{5}}{2} \\ \\ \approx -1.98416887968358328512416339196512602478420176057088655842818644\ldots$

$1\,-\,12\ln{A} \approx -1.9850537244054111505670359229133677131684365640224214\ldots$

$\zeta''(0) = -\dfrac{\ln^2(2\pi)}{2} - \dfrac{\pi^2}{24} + \dfrac{\gamma^2}{2} + \gamma_1 \\ \\ \approx -2.006356455908584851210100026729960438198994910160919881169868280\ldots$

$\zeta(1) \,-\, H_7 = \gamma \,-\, \dfrac{363}{140} \approx -2.01564147795560999653634505277474042610069780\ldots$

$\dfrac{\xi'(-1)}{\xi(-1)} - \dfrac{\zeta'(-1)}{\zeta(-1)} = -\dfrac{1}{2} (1+\gamma+\ln(4\pi)) \\ \\ \approx -2.054119955935411826792201842175907139420227208787872839280375162\ldots$

$\zeta(1) \,-\, H_8 = \gamma \,-\, \dfrac{761}{280} \approx -2.14064147795560999653634505277474042610069780\ldots$

$-\pi\ln2 \approx -2.177586090303602130500688898237613947338583700369286294325\ldots$

$\zeta(1) \,-\, H_9 = \gamma \,-\, \dfrac{7129}{2520} \approx -2.2517525890667211076474561638858515372118089\ldots$

$\dfrac{1}{e} \left(\text{Ei}(1) - \text{Ei}(e)\right) \approx -2.323733975225242847464643064350826137264218535575\ldots$

$\zeta(1) \,-\, H_{10} = \gamma \,-\, \dfrac{7381}{2520} \approx -2.3517525890667211076474561638858515372118089\ldots$

$-\dfrac{\pi^2}{4} \approx -2.46740110027233965470862274996903778382842485181019765660333\ldots$

$H_{-\frac{2}{3}} = H_\frac{2}{3} - \dfrac{3}{2} + \pi\cot(\frac{2\pi}{3}) = -\dfrac{\pi}{2\sqrt{3}} - \dfrac{3\ln 3}{2} \\ \\ \approx -2.554818115119273462389906984204866423113268960780494465386983968\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi^2}{1920} = \ln\!\dfrac{\pi}{8\sqrt{30}} \approx -2.6353103466615134418148873588249159487035016133\ldots$

$\dfrac{\xi'(\frac{1}{2})}{\xi(\frac{1}{2})} - \dfrac{\zeta'(\frac{1}{2})}{\zeta(\frac{1}{2})} = -\dfrac{\pi}{4} - \ln 2 - \dfrac{\ln(2\pi)}{2} - \dfrac{\gamma}{2} \\ \\ \approx -2.686091709612832791116478748724871144507269625811776921584451315\ldots$

$-1 \,+\, 2\gamma \,-\, 12\ln A \approx -2.830622394602345429354011742748562851084117892142\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{960} = \ln\!\dfrac{\sqrt{\pi}}{8\sqrt{15}} \approx -2.8611016993062408741779849737723570204893989526\ldots$

$5(1-\ln5) \approx -3.0471895621705018730037966661309381976280067713425886095\ldots$

$\dfrac{\zeta(2)}{\zeta(0)} = -2\pi^2 \, \dfrac{B_2}{B_0} = -\dfrac{\pi^2}{3} \approx -3.289868133696452872944830333292050378437899\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{2688} = \ln\!\dfrac{\sqrt{\pi}}{8\sqrt{42}} \approx -3.3759114078968199941388977396099413495828907075\ldots$

$\Gamma(-\frac{1}{2}) = -2\sqrt{\pi} \approx -3.544907701811032054596334966682290365595098912244\ldots$

$H_{-\frac{3}{4}} = H_\frac{3}{4} - \dfrac{4}{3} + \pi\cot(\frac{3\pi}{4}) = -\dfrac{\pi}{2} - 3\ln 2 \\ \\ \approx -3.650237868474732547483018056014281146325085102768318672849512324\ldots$

$-2\ln(2\pi) \approx -3.6757541328186909671213189456224705594455898945511336512\ldots$

$\zeta'(\frac{1}{2}) \approx -3.9226461392091517274715314467145995137303239715065052095683\ldots$

$\zeta'(\frac{3}{2}) \approx -3.93223973743110151070638857840601520269274355489257726154466\ldots$

$\Gamma(-\frac{2}{3}) \approx -4.018407802061621450483539411462016466193034066935951651425\ldots$

$\Gamma(-\frac{1}{3}) \approx -4.062353818279201250835864084463541356557981798170381095182\ldots$

$\Psi^{(0)}(\frac{1}{4}) = \dfrac{\Gamma'(\frac{1}{4})}{\Gamma(\frac{1}{4})} = H_{-\frac{3}{4}} - \gamma = -\dfrac{\pi}{2} - 3\ln 2 - \gamma \\ \\ \approx -4.22745353337626540808953014609668357736724443870824227165527955\ldots$

$H_{-\frac{4}{5}} = H_\frac{4}{5} - \dfrac{5}{4} + \pi\cot(\frac{4\pi}{5}) = -\dfrac{\pi}{2} \, \sqrt{1+\dfrac{2}{\sqrt{5}}} - \dfrac{5\ln 5}{4} - \dfrac{\sqrt{5}}{4} \ln\!\left(\dfrac{3\!+\!\sqrt{5}}{2}\right) \\ \\ \approx -4.71182423169065543494069587236754967378342893826671921137009142\ldots$

$6(1 - \ln6) \approx -4.7505568153683300048748641502842136363379441530980282351\ldots$

$\Gamma(-\frac{3}{4}) \approx -4.834146544295877749240913541156896003993556910506753956577\ldots$

$\Gamma(-\frac{1}{4}) \approx -4.901666809860710580516393213451562107404956992432282444920\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{64512} = \ln\!\dfrac{\sqrt{\pi}}{96\sqrt{7}} \approx -4.96493832307079280396236854025846905401988618\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{165888} = \ln\!\dfrac{\sqrt{\pi}}{288\sqrt{2}} \approx -5.4371691274912184975135534661884931778865844\ldots$

$H_{-\frac{5}{6}} = -\dfrac{\pi\sqrt{3}}{2} \,-\, 2\ln 2 \,-\, \dfrac{3\ln 3}{2} \approx -5.7549118404733819318184494847633752915\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi^2}{2580480} = \ln\!\dfrac{\pi}{192\sqrt{70}} \approx -6.23701310720306086831688271338229837007228\ldots$

$\dfrac{\xi(-1)}{\zeta(-1)} = -2\pi \approx -6.283185307179586476925286766559005768394338798750211\ldots$

$7(1 - \ln7) \approx -6.6213710433871931357374692041022581074595931070730283192\ldots$

$\Gamma(-\frac{5}{6}) \approx -6.679579202136282245100116274249271333678558937847424153940\ldots$

$\Gamma(-\frac{1}{6}) \approx -6.772722179448755767565406541553052079960724649988532710512\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{6635520} = \ln\!\dfrac{\sqrt{\pi}}{1152\sqrt{5}} \approx -7.28160885454818664893978131498885184976263\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi}{16220160} = \ln\!\dfrac{\sqrt{\pi}}{384\sqrt{110}} \approx -7.728517792559234884284123927777979079063\ldots$

$8(1 - \ln8) \approx -8.6355323334386874260135709149962376338120032246461260988\ldots$

$\dfrac{1}{2}\ln\!\dfrac{\pi^2}{10701766656} = \ln\!\dfrac{\pi}{27648\sqrt{14}} \approx -10.4021074505620113014759220064469622\ldots$

$9(1 - \ln9) \approx -10.775021196025974445114414264605462683654830040809490131\ldots$

$\dfrac{\zeta'(0)}{\zeta(0)} - \dfrac{\zeta'(-1)}{\zeta(-1)} = 1 + \ln(2\pi) - 12 \ln A \\ \\ \approx -12.55124848279812615894345135362626545348947218062444747913057717\ldots$

$10(1 - \ln10) \approx -13.0258509299404568401799145468436420760110148862877297\ldots$

$\dfrac{\zeta'(2)}{\zeta(2)} - \dfrac{\zeta'(-1)}{\zeta(-1)} = 1 + \ln(2\pi) + \gamma - 24 \ln A \\ \\ \approx -29.36315836710406494084105008998136375565957997258453818508969020\ldots$