Inverse Laplace transform of $\frac{\sqrt{s}}{\sinh(\sqrt{s})}$

Question

I was just wondering if it is possible to determine the Inverse Laplace Transforms of the function

$$F(s) = \frac{\sqrt{s}}{\sinh(\sqrt{s} )}$$

by utilising the Residue theorem. I am fine in finding the residues at $s=n^2 \pi^2$, however, I am having trouble finding the residue at $s=0$.

I think the question itself might be incorrect and is meant to be, instead

$$F(s) = \frac{\sqrt s}{s \sinh(\sqrt{s} )}$$

Thanks for the help.