Analytic continuation of Zeta type function

Question

Can one analytically continue the function (Not equal to the Zeta function)

$$Z(s)=\prod_{p}\frac{1}{1+p^{-s}}=\sum_{k=1}^{\infty}\frac{(-1)^{\Omega(k)}}{k^s}$$ Where $\Omega(k)$ is the number of distinct prime factors of $k$.

Specifically, can one find $Z(0)$?

Answer 1

$$Z(s)=\frac{\zeta(2s)}{\zeta(s)}$$