Don't worry. Anyway, if somebody can give me suitable references (about how to deal with this kind of zeta and how to regularise it) I would be much appreciated (in the general case, since I cannot take the $a>>1$ approx)

btw, @CarloBeenakker, how do you get the result you wrote for $p=3/2$, Mathematica can't find the explicit form...

Ok, but what about general values for $a$? Isn't $F_a(s) - \frac{1}{s-1/2}$ finite for $s \rightarrow 1/2$ ?

Do I have to expand around $\epsilon$?

Thank you guys! Now, how can I exactly regularise it by adding $\epsilon$ to $p$ (never done that, sorry for the dumb question)? I need to find a way to extract a finite result like in the Riemann's zeta case (where we can take express the zeta as div+finite parts and one can take the average to cancel the div part, for instance). Moreover, I want to keep $a$ as general as possible, without making any assumption like $a>>1,a<<1$. ( $a>>1$ has actually a physical meaning, but then I recover the standard zeta, when $p=1/2$, and I know how to regularise). I want to understand the general case.