While playing with Cohen's pari script prodeulerrat found a function.

For $s \in \mathbb{C}$ define $$ f(s) = \prod_{p \text{ prime}} (1-\frac{s^2}{p^2})$$

The product converges everywhere, no poles and the zeros are $\pm p$.

At integers one can tell if $f(n)=0$ via primality testing.

Cohen's script computes $f(s)$ in $O(|s|)$ and it iterates over primes.

Q1 Is there an alternative way to compute $f(s)$?

Q2 An explicit series converging to $f(s)$?

$f(1)=1/\zeta(2)$.

Q3 Is there closed form for $f$ at integers?

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