Inverse Laplace transform of $e^{-s^{1/2}}$

Question

Inverse Laplace transform of $e^{-s^{1/2}}$

I tried solving this question using the series expansion of $e^x$. However, I cant seem to proceed ahead with the inverse transform

Answer 1

I think you can use the fact that for $a>0$, the inverse transform of $e^{-\sqrt{ap}}$ is $\frac{\sqrt{a}}{2\sqrt{\pi}}x^{-3/2}e^{-\frac{a}{4x}}$. Please vote if you find this helpful.