A zeta function using half of the primes

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?

flag	asked Feb 8 at 3:48 Khalid Bou-Rabee 586●2●13

Every other prime seems to be an arbitrary choice. In my opinion, a more interesting question borne of the same motivation would be: for which subsets $S$ of the primes does the corresponding Euler product admit an analytic continuation? – spec Feb 8 at 4:05

Similar questions have been asked: mathoverflow.net/questions/95205/…, mathoverflow.net/questions/70318/…, mathoverflow.net/questions/28000/…, not that they answer your specific question, but maybe a good place to look... – BR Feb 8 at 4:19

Thank you. I wrote this specific question because it is appearing in my work. I agree that the more general question is also interesting. – Khalid Bou-Rabee Feb 8 at 15:30