I am reading Nigel Hitchin's paper on Harmonic spinors [Adv. Math. 14 (1974) 1-55]. On p 34 he defines a function $$ f(s)= \sum_{p,q\in{\mathbb Z}_+ }(p+q)(4pq \lambda^2 +(p-q)^2)^{-s}. $$ The series converges absolutely for $s>3/2$, and he says "Computing the residues at $s=1/2$, $s=3/2$ we finally....". I have no idea how to compute these residues. Can anyone suggest a suitable method for this and related sums?
The sum is over positive integers only, so it does not appear to be an Epstein zeta function.