I'm a string theorist and I have come across the following expression in a computation I'm doing (involving a sum over inequivalent Lens spaces): $$\widehat{\zeta}(s)=\prod_{\mathrm{primes}\ p\equiv 3\ (\mathrm{mod}\ 4)}(1-p^{-s})^{-1}.$$ I expected this to be a standard sort of construction, and it's clearly closely related to certain standard zeta-functions and Dirichlet $L$-functions (for instance it seems that $\widehat{\zeta}(s)^2/\widehat{\zeta}(2s)=L(\chi_0,s)/L(\chi_1,s)$, where $\chi_0$ and $\chi_1$ are the trivial and nontrivial Dirichlet characters modulo $4$), but with my poor knowledge of this very wide field, I wasn't quite able to find this particular combination. I would like to extend this analytically in $s$ and am particularly interested in any poles along the real axis and their residues.

Any help or references would be much appreciated, and apologies if there was an easily found answer that I missed.