It seems that

\begin{align} &\prod_{\Omega(n)=2}^{}\dfrac{1}{1 - n^{-s}}\approx\zeta (s)\exp \left(P(s)^2/2-P(s)\right)\\ \end{align}

where $P(s)$ is the prime zeta function, $\Omega(n)$ is the number of prime divisors (with mutiplicity) of $n$, and where the RHS is the dominant term in the expansion of the Euler product.

Is this close enough to be of use in any practical application?