The author proves that \(L(\tfrac{1}{2},\chi)\neq 0\) for at least \(34\) primitive characters \(\chi\) for a large modulus \(q\). In fact, he shows that \[ \mathop{\sum\nolimits^*}_{_{\substack{\chi\pmod q,\\ L(1/2,\chi)\neq 0}}} 1 \geq (0.3411+o(1)) \mathop{\sum\nolimits^*}_{\chi\pmod q} 1\qquad(q\to\infty), \] where \(\sum^*\) denotes summation over primitive characters \(\chi\) modulo \(q\). This result improves a result of H. Iwaniec and P. Sarnak [in: Number theory in progress. Volume 2, Berlin: de Gruyter, 941–952 (1999; Zbl 0929.11025)], who proved the same result with the constant \(1/3\). The main new ingredient is a new mollifier, which is explained in detail in the text. The proof is rather long and technical, as is to be expected in this problem.