Is it possible to write the following double product in terms of the zeta function?

\begin{align} &\prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}} \end{align}

Extending the zeta function to semiprimes with

\begin{align} &\prod _{i=1}^{\infty}\dfrac{1}{1 -\ q_{i}^{\ \ -s}}\\ \end{align}

where $q$ runs through the semiprimes is rewritable as

\begin{align} &\left( \prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i\ p_j)^{-s}}\right)^{1/2}\left( \prod_{i=1}^{\infty} \frac{1}{1-(p_i{^2})^{-s}}\right)^{1/2} \end{align}

The last part can clearly be rewritten as $\zeta(2s)^{1/2}$, but I don't know how I can rewrite the first part in terms of the zeta function.

In addition, I believe it may be that the triprime zeta function

\begin{align} &\prod_{\Omega(q)=3}^{}\dfrac{1}{1 -\ q^{\ -s}}\\ \end{align}

can be written

\begin{align} &\left(\prod_{i=1}^{\infty} \prod_{j=1}^{\infty}\prod_{k=1}^{\infty} \frac{1}{1-(p_i\ p_j\ p_k)^{-s}}\right)^{1/6}\left( \prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i^{\ 2}\ p_j)^{-s}}\right)^{1/2}\left( \prod_{i=1}^{\infty} \frac{1}{1-(p_i{^3})^{-s}}\right)^{1/3} \end{align}

and the function

\begin{align} &\prod_{\Omega(q)=4}^{}\dfrac{1}{1 -\ q^{\ -s}}\\ \end{align}

can be written

\begin{align} &\left(\prod_{i=1}^{\infty} \prod_{j=1}^{\infty}\prod_{k=1}^{\infty} \prod_{l=1}^{\infty}\frac{1}{1-(p_i\ p_j\ p_k\ p_l)^{-s}}\right)^{1/24} \left(\prod_{i=1}^{\infty} \prod_{j=1}^{\infty}\prod_{k=1}^{\infty} \frac{1}{1-(p_i^{\ 2}\ p_j\ p_k)^{-s}}\right)^{1/4}\\ &\left( \prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i^{\ 3}\ p_j)^{-s}}\right)^{1/3} \left( \prod_{i=1}^{\infty}\prod_{j=1}^{\infty} \frac{1}{1-(p_i^{\ 2}\ p_j^{\ 2})^{-s}}\right)^{1/8} \left( \prod_{i=1}^{\infty} \frac{1}{1-(p_i{^4})^{-s}}\right)^{1/4} \end{align}

which of course presents the greater problem of defining the further multiple products in terms of the zeta function.