If I write $$ H(x) = \frac{1}{x^2} + \frac{1}{(x+1)^2} + \ldots + \frac{1}{(x+k)^2} + \ldots $$ I can see that finding $H(1)$ amounts to the Basel problem, which makes me a bit pessimistic, but...
Is there any chance that $H$ has some "nice" representation that doesn't directly involve an infinite sum (even if some constant like $\zeta(2)$ appears)?
It's been nearly a half-century since I studied complex variables, and I wasn't great at it even back then, alas.
Perhaps I should also mention the more general problem from which I extracted this one, because it may actually be simpler. I'd like to sum up $$ L(x) = \frac{1}{x^2} + \frac{1}{(1+x)^2} + \frac{1}{(1-x)^2}\ldots + \frac{1}{(2+x)^2} + \frac{1}{(2-x)^2} + \ldots $$