It seems that

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

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?