To what extent the Laplace transform of the function $x^q$ is justified when $-1<q<0$?

Question

The (one-sided) Laplace transform of the function $x^q$ according to the tables is ${\operatorname {\Gamma } (q+1) \over s^{q+1}}$. According to the tables, it is valid for $q>-1$. Other tables though seem to exclude the non-integer cases. But my calculations show problems with $-1<q<0$ as well. Thus I wonder, to what extent the formula is justified at (non-integer) $-1<q<0$?