Does Fourier imply Laplace?

Question

Can we find a function $f(t)$ for which $$\int_{-\infty}^{+\infty}f(t)e^{-j\omega t}dt,$$ converges but $$\int_{-\infty}^{+\infty}f(t)e^{-st}dt,$$ does not ?

Here, $j^2=-1$, $\omega$ is a real number and $s$ is a complex number.

I am thinking that we can find such an $f(t)$, for example, when $Re(s)>0$.

Answer 1

For instance, $f(t)=e^{-\lvert t\rvert^{1/3}}$, because $\int_{-\infty}^0 e^{-st-\lvert t\rvert^{1/3}}\,dt$ diverges for $\Re s>0$.

Answer 2

$f(t)=\frac 1 {1+t^{2}}$ is such a function.