Let \(K/\mathbb{Q}\) be a finite, Galois extension of \(\mathbb{Q}\), with Galois group \(G= \text{Gal}(K/\mathbb{Q})\). The main purpose of this paper is to obtain a generalization of a theorem of Voronin, and others (see below and the references in the paper) on the distribution of the values of the Riemann zeta-function and Dirichlet \(L\)-series to the case of the Artin \(L\)-functions \(L(s,\chi_j,K/\mathbb{Q})\) associated with a set of linearly independent characters \(\chi_1,\dots, \chi_n\) of \(G\). More precisely, the author proves the following theorem, which is the principal result of this paper and whose many applications are explained and developed in it. With the foregoing notation, denote by \(k\) the order of \(G\) and let \(f_1(s),\dots, f_n(s)\) be holomorphic functions on the set of \(s\in\mathbb{C}\) such that \(| s| 0\), there is a set \(A_\varepsilon\subset \mathbb{R}\) such that \[ \liminf_{T\to\infty}\, {\text{vol} (A_\varepsilon \cap (0,T))\over T}> 0, \] and for \(j= 1,2,\dots, n\) and any \(t\in A_\varepsilon\) and \(| s|\leq r\), \[ \Biggl| L\Biggl(s+ 1-{1\over 4k}+ it,\,\chi_j, K/\mathbb{Q} \Biggr)\Biggr| 1\), which states that those zeta-functions have zeros with \(\text{Re}(s)> 1\). Voronin [see \textit{A. A. Karatsuba} and \textit{S. M. Voronin}, The Riemann zeta-function (de Gruyter Expositions in Mathematics 5, de Gruyter, Berlin) (1992; Zbl 0756.11022)] proved that those functions do also have zeros in the strip \({1\over 2}1\). More precisely, he proves the following theorem. Denote by \(O_K\) the ring of integers of the number field \(K\) and by \(H_{\mathfrak f}\) an ideal group with conductor \({\mathfrak f}\); let \(I^{({\mathfrak f})}\) denote the group of fractional ideals of \(O_K\), prime to \({\mathfrak f}\). Then, if \(I^{({\mathfrak f})}/H_{\mathfrak f}\) contains more than one ideal class, the partial zeta-function \(\zeta(s,{\mathcal A})\) attached to any class \({\mathcal A}\in I^{({\mathfrak f})}/H_{\mathfrak f}\) has infinitely many zeros in the strip \({1\over 2} 0\) is sufficiently large, then there exists \(c> 0\) such that there are at least \(cT\) zeros of \(\zeta(s,{\mathcal A})\) in the region defined by \({1\over 2}< \text{Re}(s)< 1\) and \(|\text{Im}(s)|< T\). Those very interesting deductions from the main theorem are developed in the course of the paper, though the proof of the main theorem is complicated and draws on many ideas, which the author sets out very lucidly in the paper.