It's trivial that the Laplace Transform of a positive function is a positive function on $s$ domain. What about the inverse thought? What can we say about the positiveness of the inverse Laplace Transform of a positive function of form $s^{-\alpha}$?

The answer is trivial when $\alpha>0$, because $\mathcal{L}\{t^{\alpha-1}\}=\Gamma(\alpha) s^{-\alpha}$ when $\alpha>0$.

But and if $\alpha<0$? The exact function is not known by any people, but could we say something about positiveness?

Thank you so much.