The \(\tau \)-Keiper/Li coefficients attached to a function F are closely related to its zero-free regions. However, the absence of a closed formula for calculating these coefficients makes them challenging to use. Motivated by Voros approach, we introduce the discretized \(\tau \)-Keiper/Li coefficients. A finite sum representation derived for these coefficients is useful for numerical calculations. Representation in terms of zeros of the corresponding function is basis for analytic considerations. We prove that the violation of \(\tau /2\)-generalized Riemann hypothesis implies oscillations of corresponding discretized \(\tau \)-Li coefficients with power-growing amplitudes. Results are obtained for the class \({\mathcal {S}}^{\sharp \flat }(\sigma _0, \sigma _1)\), which contains the Selberg class, the class of all automorphic L-functions, the Rankin–Selberg L-functions, as well as products of suitable shifts of those functions.

附加到函数F的\（tau \）- Keiper / Li系数与其零自由区密切相关。但是，由于缺少用于计算这些系数的封闭公式，因此很难使用它们。受Voros方法的启发，我们介绍了离散化的\（\ tau \）- Keiper / Li系数。为这些系数导出的有限和表示对于数值计算很有用。相应函数的零表示形式是进行分析考虑的基础。我们证明违反了\（\ tau / 2 \）-广义Riemann假设意味着对应的离散化\（\ tau \）的振荡-Li系数具有随功率增长的幅度。获得类\（{\ mathcal {S}} ^ {\ sharp \ flat}（\ sigma _0，\ sigma _1）\）的结果，该类包含Selberg类，所有自胚L函数的类， Rankin–Selberg L-函数，以及这些函数的适当移位的乘积。