Order of Pole at $s=0$ while using Residue Theorem to Invert Laplace Transform

Question

I'm trying to invert the following Laplace Transform using the Residue Theorem:

$$ F(s) = \frac{K_1(\sqrt s)}{\sqrt s K_0(\sqrt s)} $$

where $K_0()$ and $K_1()$ are the Modified Bessel Functions of the Second Kind of Orders $0$ and $1$, respectively.

There is a single pole at $s=0$. However if I treat this as a first order pole (which I'm quite sure it isn't) then using the Cauchy Residue Theorem, the inverse transform evaluates to zero. This would also be true if it is regarded as a pole of any order $\ge$ 1.

Could somebody explain a way forward? Thanks.