I'm looking for the residues of the following function $$s \mapsto\sum_{p,q > 0} (p+q) \left[ 4pqa^2 + (p-q)^2 \right]^{-s}$$ at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real (positive) number.

Yet, I have literally no idea how to precedure here. I tried to rearrange the terms in order to get some well known zeta function but it failed. Is there any useful integral representation or any simple trick I'm not aware of?

I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$ at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive number.

Yet, I have literally no idea how to precedure here. I tried to rearrange the terms in order to get some well known zeta function but it failed. Is there any useful integral representation or any simple trick I'm not aware of?

Thanks in advance!

I'm looking for the residues of the following function $$\sum_{p,q > 0} (p+q) \left[ 4pqa^2 + (p-q)^2 \right]^{-s}$$ at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real (positive) number.

Yet, I have literally no idea how to precedure here. I tried to rearrange the terms in order to get some well known zeta function but it failed. Is there any useful integral representation or any simple trick I'm not aware of?

I'm looking for the residues of the following function $$s \mapsto\sum_{p,q > 0} (p+q) \left[ 4pqa^2 + (p-q)^2 \right]^{-s}$$ at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real (positive) number.

Yet, I have literally no idea how to precedure here. I tried to rearrange the terms in order to get some well known zeta function but it failed. Is there any useful integral representation or any simple trick I'm not aware of?