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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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Caleb Stanford
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    What do you mean by "standard function"? For example, for me $\zeta$ is a standard function, is it for you? – Wojowu Aug 30 '15 at 21:23
  • @Wojowu I mean something very broad. Of course $\zeta$ function is standard. Any other function that has been defined in the literature or in a paper is standard. – Caleb Stanford Aug 30 '15 at 21:29
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    See this MathOverflow question here: https://mathoverflow.net/questions/28000/what-are-the-analytic-properties-of-dirichlet-euler-products-restricted-to-arith – Peter Humphries Aug 31 '15 at 19:57
  • You might want to look up Dirichlet L functions. – TravorLZH Nov 14 '21 at 08:10

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