Laplace transform of the tetration (integral or series)

How to get some insight in the following integral:

$$$$\mathcal{I}(s)=\int_0^\infty x^{-x}e^{sx}\text{d} x$$$$

where $$s$$ is real (and the lower integration bound may be set to $$a>0$$)?

Alternatively, can we estimate this kind of series:

$$$$\mathcal{S}(s)=\sum_{n=1}^\infty n^{-n}e^{sn}$$$$ where $$s$$ is real?

• Find some discussion here: mathoverflow.net/questions/109160/… Jan 7 '19 at 12:59
• Of course we can estimate it. What kind of insight are you looking for? Jan 7 '19 at 13:21

G. H. Hardy, On the integral function $$\Phi_{a,\alpha,\beta}(z)= \sum\frac{x^n}{(n+a)^{\alpha n+\beta}},$$ Quarterly J. Math., 5 (1906) 37, 369-378. (Collected papers of G. H.Hardy, vol. IV, p. 128).