Say I have a function $f: \mathbb{R}_+ \rightarrow \mathbb{R}_+$ such that $e^{\lambda t }f(t) \in L^1(\mathbb{R}_+)$ for some $\lambda \in \mathbb{R}$. I want to compute numerically an approximation its Laplace transform: $$ \forall Re(z) \geq -\lambda, ~\hat{f}(z) = \int_0^\infty{e^{-z t} f(t) dt}. $$
My goal is then to plot the function $z \mapsto \hat{f}(z)$ or to compute its zeros so I would like to evaluate the Laplace transform approximation at multiple inputs.
Example: $f(t) := e^{1/2 (3 + e^{-2 t} - 4 e^{-t} - 2 t)}$. What are the classical algorithms to do that?
Thanks!