Let \(\psi \) be a real primitive character modulo D. If the L-function \(L(s,\psi )\) has a real zero close to \(s=1\), known as a Landau–Siegel zero, then we say the character \(\psi \) is exceptional. Under the hypothesis that such exceptional characters exist, we prove that at least fifty percent of the central values \(L(1/2,\chi )\) of the Dirichlet L-functions \(L(s,\chi )\) are nonzero, where \(\chi \) ranges over primitive characters modulo q and q is a large prime of size \(D^{O(1)}\). Under the same hypothesis we also show that, for almost all \(\chi \), the function \(L(s,\chi )\) has at most a simple zero at \(s = 1/2\).

让\（\ PSI \）是一个真正的原始字符模d。如果L函数\（L（s，\ psi）\）具有接近于\（s = 1 \）的实零，即Landau–Siegel零，那么我们说字符\（\ psi \）为例外。在存在这样的特殊字符的假设下，我们证明Dirichlet L函数\（L（s，\ chi）\）的中心值\（L（1/2，\ chi）\）的至少百分之五十是非零值，其中\（\ chi \）范围为原始字符模q，q是大小为\（D ^ {O（1）} \）的大素数。在相同的假设下，我们还表明，对于几乎所有\（\ chi \），函数\（L（s，\ chi）\）在\（s = 1/2 \）处最多只有一个零。