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)}
\quad
\underbrace{\prod_{p \equiv 3 \pmod{4}} \frac{1}{1 – p^{-s}}}_{\zeta_1(s)},
\end{align*}
so I’m asking for an evaluation of $\zeta_0(s)$ and $\zeta_1(s)$.

I encountered this while finding an expression for the probability that a “random” integer can be written as a sum of two squares, in this answer.
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
$$
\lim_{x \to 1} \frac{\zeta_1(2x)}{\zeta_1(x)}.
$$

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