I recently came across the following product over prime numbers, but lost the source of the formula. Can someone enlighten me? The product, taken over all primes $p$, is:
$$ \prod _{p}^{\infty} \frac{p^2 \left(1-p+p^2\right)}{1-p+p^2-p^3+p^4}. $$
Incidentally, Mathematica tells me that it is approximately equal to:
$$ \frac{1}{34} \sqrt{\frac{1}{6} \left(19 \sqrt{237673}-239\right)}. $$
Note that $$ \frac{p^2(1-p+p^2)}{1-p+p^2-p^3+p^4} = \frac{1+1/p^3}{1+1/p^5}$$ Now use the Euler products $$ \prod_p (1 + 1/p^s) = \frac{\zeta(s)}{\zeta(2s)}$$ so your infinite product is $$ \frac{\zeta(3) \zeta(10)}{\zeta(6) \zeta(5)}$$