Evaluating the product $\prod_{p}\, \left(1- \dfrac{2}{p^s}\right)^{-1}\,\left(1 - \dfrac{1}{p^s} \right)^2\,$

Question

I need to evaluate $$ f(s) = \prod_{p}\, \left(1- \dfrac{2}{p^s}\right)^{-1}\,\left(1 - \dfrac{1}{p^s} \right)^2\,.$$ This arises from $$\sum_{n=1}^{\infty} \dfrac{2^{\Omega(n)}}{n^s} = \zeta^2 (s) \, \prod_p\, \left(1- \dfrac{2}{p^s}\right)^{-1}\,\left(1 - \dfrac{1}{p^s} \right)^2, \, \, \, \mathscr{R}(s)>1$$

What I know is that this has a pole of order 3 at $s=1$. Then, the idea is to find the Laurent expansion of $f(s)$ and some sort of an integral formula (Perron's formula?) to find $f(s)$? Any help on how to approach this?

Answer 1

What do you mean with "evaluate" ? $$\log f(s) = \sum_{p^k} \frac{2^k - 2 }k p^{-sk}$$ converges and is analytic for $\Re(s) > 1/2$.

$f(s)$ has a simple pole at $s=1/2$.