I am a string theorist who has encountered the following number theory problem in my research.

Consider the sums

$$Z_+(s) = \sum_{p=1}^\infty p^{-s} S(1,1; p)$$

and

$$Z_-(s) = \sum_{p=1}^\infty p^{-s} S(1,-1; p)$$

where $S(n,m;p)$ is the Kloosterman sum. This sum converges when $s$ has a large real part and can be analytically continued to define $Z_{+/-}(s)$ on the whole complex $s$ plane. This is a special case of the Kloosterman zeta function; it contains data about the spectra of certain Riemann surfaces.

I am interested in the behaviour of this analytically continued function when s is an integer. In particular I would like to know whether $Z_{+/-}(s)$ has poles for any integer values of $s$, and if so what the residues are.

I've found some literature on the subject of Kloosterman Zeta functions
but I have not found quite what I'm looking for. Most treatments focus on the
general properties of the Kloosterman Zeta function, which are quite complicated.

My suspicion is that
because I am only interested in integer values the answer might be relatively simple.

Any help would be appreciated, either direct answers or specific pointers to the literature.