Take the Hadamard product for the Riemann $\xi$-function ($\rho$ is a non-trivial zero of $\zeta(s)$):
$$\xi(s) =\frac12\, s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)\, \zeta(s) =\frac12\,\prod_\rho \left(1- \frac{s}{\rho} \right) \left(1- \frac{s}{1-\rho} \right)$$
Firstly strip out the factor $\frac12$ on both sides and then find the solution for:
$$s\,(s-1) \,\pi^{-\frac{s}{2}}\, \Gamma\left(\frac{s}{2}\right)=1$$ This yields the unique value of $\mu=3.171242735975879847\dots$ and therefore:
$$\zeta(\mu) = \prod_\rho \left(1- \frac{\mu}{\rho} \right) \left(1- \frac{\mu}{1-\rho} \right)$$ Hence, since $\mu \gt 1$ this gives a direct relation between the products of primes and $\rho$'s:
$$\prod_{p \in \mathbb{P}} \left( \dfrac{1}{1-p^{-\mu}} \right)=\prod_\rho \left(1- \frac{\mu}{\rho} \right) \left(1- \frac{\mu}{1-\rho} \right)$$
Is there anything known about $\mu$? (I tried link to known constants on Wolfram, but no success)