If we let $P(s)=\sum_{p}\frac{1}{p^s}$ be the prime zeta function and $P^*(s)=\sum_{p^k}\frac{1}{p^{ks}}$ be the "prime power zeta function", then are there any known/trivial zero free regions of $P(s)$ or $P^*(s)$? In this Math.SE question the answerer states that it is not known whether $P(s)$ has any zeros with $\Re(s)=1$. Since $1$ is not a prime numbe, neither $P(s)$ nor $P^*(s)$ have Dirichlet inverses and so $\frac{1}{P(s)}$ and $\frac{1}{P^*(s)}$ have no Dirichlet series and thus cannot be shown to not have poles for $\Re(s)>1$ in a trivial way like can be done for $\zeta(s)$.
I ask this question since I have found a reasonable route of attack for showing that $P^*(s)$ has no zeros for $\Re(s)>\sigma_0$ for some fixed $\sigma_0>1$, but I don't want to carry through with the whole proof if I am missing anything trivial. Any insights would be appreciated.