Following Isometry group of an integer I wonder if one can define a "mock zeta function" $\zeta_{V}$ (where $V:=(\mathbb{Z}/2\mathbb{Z})^{2}$ stands for "Vierergruppe", the German word for the Klein group) whose non-trivial zeros would give rise to the analogue of Riemann's explicit formula (relating the non-trivial zeros of $\zeta_{\mathbb{Z}/2\mathbb{Z}}:=\zeta$ to $\pi(x)=\pi_{\mathbb{Z}/2\mathbb{Z}}(x)$) for the counting function of the semi primes $\pi_{V}(x)$. If so could the analogue of the Riemann hypothesis for $\zeta_{V}$ be proven false?
I suspect there might be a relationship between the isometry group of the multi set of non trivial zeros of $\zeta_{G}$ for $G\in\{V,\mathbb{Z}/2\mathbb{Z}\}$ (this isometry group being equal to $V$ if and only the relevant RH is false) and the isometry group of the integers as defined in the link above counted by $\pi_{G}(x)$.
To state it differently, would the so-called parity problem be overcome by a proof of RH?
