Accepting, for argument's sake, that it is possible to create a function (the opposing view is that one discovers mathematical objects, rather than creating them), you have created the function simply by writing down that formula. Well, you have created it for real part of $s$ exceeding one-half, which is where the series converges. Knowing "exact values" (again, a term we can argue over, but I'll take it to mean a finite expression in terms of well-known constants, and leave it at that), well, that's generally very difficult. No one knows an exact value for $\zeta(3)$, for example, and for your function I suspect no one knows an exact value for $F(1)$. And as to extending your function to be defined for all complex $s$ (outside of a pole at $s=1/2$), that's what analytic continuation is for, so now you have a keyphrase to search for.

Accepting, for argument's sake, that it is possible to create a function (the opposing view is that one discovers mathematical objects, rather than creating them), you have created the function simply by writing down that formula. Well, you have created it for real part of $s$ exceeding one-half, which is where the series converges. Knowing "exact values" (again, a term we can argue over, but I'll take it to mean a finite expression in terms of well-known constants, and leave it at that), well, that's generally very difficult. No one knows an exact value for $\zeta(3)$, for example, and for your function I suspect no one knows an exact value for $F(2)$. And as to extending your function to be defined for all complex $s$ (outside of a pole at $s=1/2$), that's what analytic continuation is for, so now you have a keyphrase to search for.