On the zeros of the Riemann Zeta function

Question

Let us consider the infinite set of non-trivial zeros of Riemann zeta function $\{\rho_n \}$ and the following product

$$ \prod_{n=1}^{\infty}\bigg(1-\frac{1}{\rho_n}\bigg)\bigg(1-\frac{1}{\overline{\rho}_n}\bigg)=1. $$

We trivially note that for every $n\in \mathbb{N}$, we have $$ \bigg(1-\frac{1}{\rho_n}\bigg)\bigg(1-\frac{1}{\overline{\rho}_n}\bigg) = \bigg(1-\frac{1}{1-\rho_n}\bigg)\bigg(1-\frac{1}{1-\overline{\rho}_n}\bigg) $$

if and only if $\Re(\rho_n)=1/2$. Does this mean that

$$ \prod_{n=1}^{\infty}\bigg(1-\frac{1}{1-\rho_n}\bigg)\bigg(1-\frac{1}{1-\overline{\rho}_n}\bigg) = 1 $$

if and only if $\Re(\rho_n)=1/2$, and furthermore that $\zeta(1-z)=0$ if and only if $\Re(z)=1/2$ ?

Answer 1

$$(1-\frac1s)(1-\frac1{1-s}) = \frac{s-1}{s}\frac{-s}{1-s}=1$$ no need that $\Re(s)=1/2$