Laplace transform vanishing at all integers I am looking to find a function $f$ which is  not zero almost everywhere ($f\neq 0 ~a.e.$) and such that $$ \int_{\mathbb{R}} f(t) e^{-ts} dt = 0~\forall s\in\mathbb{Z}$$
I was thinking of taking an inverse Laplace transform of sine to help me with this, but apparently it does not exist. I am not thinking it should be a more elaborate construction, perhaps some sort of piecewise function.

The OP made it more or less clear that we are dealing with the bilateral Laplace transform.
It follows from the Cauchy integral theorem that $$\int_{-\infty}^\infty e^{-(t+2i\pi)^2} e^{-st}dt=\int_{-\infty}^\infty e^{-t^2} e^{-s(t-2i\pi)}dt$$
whence take $$f(t)=e^{-t^2}-e^{-(t+2i\pi)^2}$$
