Consider the series: $$\sum\limits_{n=0}^{\infty}(-1)^nn!$$ Well know that if p is prime, this series convergence in $\mathbb{Q}_p$. Let $s_p\in\mathbb{Q}_p$ is sum of this series.

Also let $$s_{\infty}=\int\limits_{0}^{\infty}\dfrac{e^{-x}}{x+1}dx\in\mathbb{R}$$

In $\mathbb{R}$ the series is not convergence; but we are using the theory of divergent series.

My question: What is the connection between numbers $s_p$?

I want the relationship type $\prod_{p}\left(|s_p|_p\right)^{(1-1/p^3)}=1$.