I have a function $f \in L^1(\mathbb{R}_+, \mathbb{R})$ with Laplace transform $$ \forall \Re(z) \geq 0,~~ \hat{f}(z) := \int_{\mathbb{R}_+} { f(t) e^{-zt } dt}.$$
I know explicitly the expression of $f$. I would like to compute numerically (an approximation of) the complex roots of $\hat{f}$, say the first 10 roots with the smallest real part. I want an effective algorithm (I do not require mathematical guarantees on the correctness of the roots!)
How should I do that?
The assumptions here are that $f$ is smooth, $f(t)$ (and its derivatives) fast decay to zero as $t$ goes to infinity and I can compute numerically $f(t)$ for a fix $t$ quite fast.
What I tried is to first compute an approximation of $f$ is a weighted Laguerre polynomial basis - for which the Laplace transform is known explicitly - and then compute the roots using contour integrations on rectangles. It works on toy examples but it is not very robust... So I would like to avoid to compute $\hat{f}$, if possible.
There is no guarantee that there are "$10$ roots with the smallest real part". That is, it is quite possible that there is an infinite sequence of roots with real parts positive but approaching $0$, and no roots with real part $0$.