Given the standard unilateral Laplace transform defined on $L^1(\mathbb R ^+)$ $ \mathscr Lf(s) = \int_0^\infty e^{-st}f(t)~dt,$
are there any functions in $L^1$ such that $\mathscr Lf$ is "compactly supported", where with compact support I mean that there exists an $M >0$ such that $\mathscr Lf(s) = 0 \quad\text{if}\ \operatorname{Re} s > M.$