Divergent Products over non-trivial zeros of the Riemann zeta function

List of divergent infinite products over the non-trivial zeros $\rho$ of the Riemann $\zeta(\,)$ (with $\rho$ and $1-\rho$ paired together).

$\zeta(\,)$ denotes the Riemann zeta function.
$\xi(\,)$ denotes the Riemann xi function.
$\rho$ denotes the non-trivial zeros of the Riemann zeta function.
$s \; (s \in \mathbb{C})$ denotes a complex variable.

SECTION

$\displaystyle \prod_{\rho} s = \dfrac{1}{\sqrt{s}}$

$\displaystyle \prod_{\rho} (1-s) = \dfrac{1}{\sqrt{1-s}}$

$\displaystyle \prod_{\rho} (1+s) = \dfrac{1}{\sqrt{1+s}}$

$\displaystyle \prod_{\rho} \left(1 - \dfrac{1}{\rho}\right) s = \dfrac{1}{\sqrt{s}}$

$\displaystyle \prod_{\rho} (\rho - 1) s = \prod_{\rho} (\rho s - s) = \dfrac{1}{2\sqrt{s}}$

$\displaystyle \prod_{\rho} (\rho + 1) s = \prod_{\rho} (\rho s + s) = \dfrac{\xi(-1)}{\sqrt{s}} = \dfrac{\xi(2)}{\sqrt{s}} = \dfrac{\pi}{6\sqrt{s}}$

$\displaystyle \prod_{\rho} (\rho - s) = \xi(s)$

$\displaystyle \prod_{\rho} (\rho + s) = \xi(-s) = \xi(1+s)$

$\displaystyle \prod_{\rho} \left(\frac{\rho}{s} - 1\right) = \sqrt{s} \; \xi(s)$

$\displaystyle \prod_{\rho} \left(\frac{\rho}{s} + 1\right) = \sqrt{s} \; \xi(-s) = \sqrt{s} \; \xi(1+s)$

$\displaystyle \prod_{\rho} \left(\rho - \dfrac{1}{s}\right) = \xi\left(\dfrac{1}{s}\right)$

$\displaystyle \prod_{\rho} \left(\rho + \dfrac{1}{s}\right) = \xi\left(-\dfrac{1}{s}\right) = \xi\left(\dfrac{1+s}{s}\right)$

$\displaystyle \prod_{\rho} (\rho - s) \left(\rho - \frac{1}{s}\right) = \prod_{\rho} \left(\rho^2 + 1 - \rho\left(s + \frac{1}{s}\right)\right) = \xi(s) \; \xi\left(\frac{1}{s}\right)$

$\displaystyle \prod_{\rho} (\rho - s) (\rho + s) = \prod_{\rho} (\rho^2 - s^2) = \xi(s) \, \xi(-s) = \xi(s) \, \xi(1+s)$

$\displaystyle \prod_{\rho} (\rho - s) \left(1 - \frac{1}{\rho}\right) = \prod_{\rho} \left((\rho - 1) - \left(1 - \frac{1}{\rho}\right) \, s \right) = \xi(s)$

$\displaystyle \prod_{\rho} \left(\rho - \frac{1}{1-s}\right) = \xi\left(\frac{1}{1-s}\right)$

$\displaystyle \prod_{\rho} \left(\rho (1-s) - 1\right) = \prod_{\rho} (\rho - 1 - \rho s) = \dfrac{1}{\sqrt{1-s}} \; \xi\left(\frac{1}{1-s}\right)$

$\displaystyle \prod_{\rho} \left(\rho - \frac{1}{1+s}\right) = \xi\left(\frac{1}{1+s}\right)$

$\displaystyle \prod_{\rho} \left(\rho (1+s) - 1\right) = \prod_{\rho} (\rho - 1 + \rho s) = \dfrac{1}{\sqrt{1+s}} \; \xi\left(\frac{1}{1+s}\right)$

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