For \(k\) a positive integer, and \(\chi\) a Dirichlet character modulo \(k,\) let \(L_\chi(s)\) denote the corresponding Dirichlet \(L\)-function. For \(a \in \mathbb{C}\) the zeros of \(L_\chi - a\) are called the \(a\)-points of \(L_\chi\), and the \textit{non-trivial} \(a\)-points are those that lie in the strip \(0 < \mathrm{Re}(s) < A\) for some \(A\) that depends on \(a\). For the Riemann zeta-function \(\zeta(s)\), \textit{A. Selberg} [in: Proceedings of the Amalfi conference on analytic number theory, Maiori, Amalfi, Italy, 1989. Salerno: Universitá di Salerno, 367--385 (1992; Zbl 0787.11037)] conjectured that three fourths of the \(a\)-points are to the left of \(\mathrm{Re}(s)=1/2\). The paper under reviews concerns itself with studying the multiplicity of the \(a\)-points. The main result is Theorem. For \(a \in \mathbb{C}\), there exists a positive percentage of the simple \(a\)-points of \(\zeta(z)\) except for at most two values. (The results also extend to Dirichlet \(L\)-functions.) The key idea of the proof is an application of Nevanlinna theory. A second result comes from the anonymous referee for this paper: Theorem. Let \(a_1,a_2,a_3 \in \mathbb{C}\) be distinct. Then for at least one of \(a_1,a_2,a_3\) the proportion of \(a_j\)-points of \(\zeta(s)\) which are simple exceeds \(\tfrac{1}{3} -\varepsilon\).