Functional equation for $\sum_{n=1}^{\infty} \sinh(cn)^{-s}$?

Does anyone know of any kind of functional equation (or closed form) for $\sum\limits_{n=1}^{\infty}\sinh(cn)^{-s}$, where $c$ is an arbitrary constant? I've been messing around with it off and on for a while now, but haven't been able to come up with anything particularly nice. Any ideas or references where such series are considered would be appreciated. Thank you in advance!

• It seems that if $s$ is an integer, the closed form could express as a linear combination of some ugly q-digamma functions. – Claude Leibovici Nov 21 '15 at 5:08
• @ClaudeLeibovici Yeah, I've tried writing out what a general form expressed that way (at least for $s\in\mathbb{Z}$) would look like, but it always ends up being something really, really terrible and entirely an extrapolation of things WolframAlpha gives me. – Ben Sheller Nov 21 '15 at 5:18