On the behavior of $p$-adic Euler $\mathrm{\ell }$-functions

In this paper we propose a construction of $p$-adic Euler $\mathrm{\ell }$-function using Kubota-Leopoldt’s approach and Washington’s one. We also compute the derivative of $p$-adic Euler $\mathrm{\ell }$-function at $s=0$ and the values of $p$-adic Euler $\mathrm{\ell }$-function at positives integers.