When working on certain integral transforms I have come across the following unusual looking series;

$$ E(z_1,z_2,s) = \sum_{ \{ \ (a,b,c,d)\in \mathbb{Z}^4 \ | \ ad-bc=1 \ \} } \frac{ \Im(z_1)^s\Im(z_2)^s}{ |(az_1+b)\bar{z}_2+(cz_1+d)|^{2s}} $$ where $\Im(z_1),\Im(z_2)>0$, and $\Re(s)>>0$.

I would be very interested i knowing whether this function is known to have a meromorphic continuation in $s$, and - assuming such a meromorphic continuation exists - whether there exist a way of evaluating this function to high precision outside the half-plane where the above series converges.

More generally I would be interested in knowing whether this function is known in the literature, and if so under which name.

UPDATE: Let me say a few words about what I would like to do with these functions. Setting $$ E^*(z_1,z_2,s) = \Gamma(s) E(z_1,z_2,s)$$ my goal would be to make sense of $E^*(z_1,z_2,-n)$ where $n$ is any positive integer, preferably by expressing it as a sum of simpler/more well-known functions. Ideally this would enable me to make sense of the integral $$ R.N. \int_0^{\infty} E^*(z_1,iy,-n) y^m \frac{dy}{y}$$

A quick calculation where I completely ignore any question of convergence seems to indicate that the integral given above might be expressible as a finite sum of functions of the form $$P_k(z,s) = \sum_{n\in \mathbb{Z}} \sum_{\gamma} \frac{\Im(\gamma.z)^s}{n^k} e^{2\pi i n \gamma.z}$$ where we are summing over $\gamma\in \operatorname{SL}_2(\mathbb{Z})$. This is done by rewriting the original sum defining $E(z_1,z_2,s)$ as a sum over $K$-Bessel functions, and then using that $K_{n-1/2}$ is elementary. Sadly I can't seem to turn this line of reasoning into anything more than a purely formal argument, hence it may very well be completely nonsensical. My hope was that either the function $E(z_1,z_2,s)$ was known in the literature and that people had studied it and related it to other more well-known objects, or that there existed a rapidly converging expression allowing me to at least numerically test the extent to which the above formal argument might have some merit.

I don't know whether my updated remark shows anything other than my own blind optimism when it comes to formal arguments. Let me close by emphsizing that I would be very interested in any thoughts on the original question, regardless of whether they may or may not relate to the updated remark.