There are no restrictions on zeros of such functions $Lf$ (unless you somehow restrict your class $C$ of functions $f$). For example, let $f$ be an infinite sum of $\delta$-functions sitting at integers. Then your integral becomes a sum, and changing the variable to $z=e^{-s}$ we obtain an arbitrary Laurent series in $\mathbf{C}\backslash\{0\}$. This can have arbitrary sequence of zeros.
The main condition of Lerch's theorem is that $f(x)=0$ for $x<0$, which ensures that $LF$ is bounded in the right half-plane. If we allow an arbitrary support of $f$, no conclusion about zeros can be made, except, of course that this is a discrete set.