Sur les formules explicites de théorie des nombres

"Explicit formulas" in number theory express the sum of values of a function at the zeros of an $L$-series of an algebraic number field as a sum of local terms corresponding to all norms of this field. For the case of Hecke $L$-series, such formulas were found earlier by the author. In this paper they are derived for Artin–Hecke series corresponding to arbitrary finite-dimensional representations of the group $W_{k,K}$ – the standard extension of the idele class group of a global field $K$ using the Galois group of a finite extension $K/k$. In this connection it turns out that terms corresponding to archimedean and nonarchimedean norms can be written in unique form.