# Riemann zeta function Euler product for primes equivalent to $3$ mod $4$

Question: can
$$\zeta_1(s) = \prod_{p \equiv 3 \pmod{4}} \frac{1}{1 – p^{-s}}$$
be evaluated or written in terms of standard functions?

Details:

We can write the Riemann zeta function as
\begin{align*}
\zeta(s)
&= \prod_{p \text{ prime}} \frac{1}{1 – p^{-s}} \\
&= \frac{1}{1-2^{-s}}
\;\;
\underbrace{\prod_{p \equiv 1 \pmod{4}} \frac{1}{1 – p^{-s}}}_{\zeta_0(s)}
so I’m asking for an evaluation of $\zeta_0(s)$ and $\zeta_1(s)$.
Since an integer is the sum of two squares iff it has an even number of each prime factor $\equiv 3 \pmod{4}$, the answer probability can be naturally written (after some work) as
$$\lim_{x \to 1} \frac{\zeta_1(2x)}{\zeta_1(x)}.$$