Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}[\sqrt t] = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} exist?

Here are a few things I've considered in trying to answer this question myself:

1. $\mathscr{L}[t^p] = \frac{\Gamma (p+1)}{s^{p+1}}$, provided that $p>-1$. The latter condition precludes the case of interest, which would correspond to $p=-3/2$.

2. The square root function can be written as $ \sqrt s = e^{\frac{1}{2}\log s}$. So $\sqrt s$ is not holomorphic at the origin.

3. This topic has been discussed previously here without clear resolution.

4. Mathematica throws the transform back at me without a solution.

Does the inverse Laplace transform, defined by the integral, \begin{equation} F(t) = \mathscr L_s^{-1}\left[\sqrt s\right](t) = \int_{c - i\infty}^{c + i\infty} \sqrt s ~e^{-st} ds \end{equation} exist?

Here are a few things I've considered in trying to answer this question myself:

1. $\mathscr{L}[t^p] = \frac{\Gamma (p+1)}{s^{p+1}}$, provided that $p>-1$. The latter condition precludes the case of interest, which would correspond to $p=-3/2$.

2. The square root function can be written as $ \sqrt s = e^{\frac{1}{2}\log s}$. So $\sqrt s$ is not holomorphic at the origin.

3. This topic has been discussed previously here without clear resolution.

4. Mathematica throws the transform back at me without a solution.