It is well known that the *zeta function* satisfies the *Euler product formula*. See this wikipedia article.

Enumerate all primes by $p_1, p_2, \ldots $ in ascending order. Set $S$ to be the set of all $p_i$ where $i$ is odd. If, for $s > 1$, you define $$\zeta_0(s) = \prod_{p \in S} \frac{1}{1-p^{-s}},$$ is it true that $(\zeta_0(s))^2$ has a meromorphic continuation to the entire complex plane?