Convolution using the Laplace integral transform of certain functions

Question

I am trying to convolve two functions:

$f(t) = e^{- t}$

$g(t) = e^{-(e^{-t})^2}$

$(f*g)(t) = \int_{0}^{t} f(t-\tau)g(\tau) d\tau = \int_{0}^{t} e^{-(t-\tau)} e^{-(e^{-\tau})^2} d\tau$

Using the convolution property, we can transform it into a product of folding functions, each of which is obtained using the direct Laplace transform of each of the original functions, i.e.:

$(f*g)(t) \rightarrow L^{-1}(L(f(t)) \cdot L(g(t)))$

But it is known that the original functions must meet certain requirements.

With these functions, the Laplace transform will look like:

ut = Exp[-Exp[-t]^2]
LaplaceTransform[ut, t, s] LaplaceTransform[Exp[-t], t, s]

And this is result:

$\frac{\Gamma \left(\frac{s}{2}\right)-\Gamma \left(\frac{s}{2},1\right)}{2 s+2}$

InverseLaplaceTransform[(Gamma[s/2] - Gamma[s/2, 1])/(
     2 + 2 s), s, t]
Out[85]= InverseLaplaceTransform[(Gamma[s/2] - Gamma[s/2, 1])/(
 2 + 2 s), s, t]

But the inverse Laplace transform of this function does not work. What could be the reason and are there any ways to find the inverse Laplace transform of this function?

I will be glad to any advice and help.