Let $L(s,\chi )$ be the Dirichlet L-function associated to a non-principal primitive Dirichlet character χ defined $\mathrm{mod}q$, where q is an odd prime. In this paper we introduce a fast method to compute $|L(1,\chi )|$ using the values of Euler's Γ function. We also introduce an alternative way of computing $\log \mathrm{\Gamma }(x)$ and $\psi (x)={\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }(x)$, $x\in (0,1)$. Using such algorithms we numerically verify the classical Littlewood bounds and the recent Lamzouri-Li-Soundararajan estimates on $|L(1,\chi )|$, where χ runs over the non-principal primitive Dirichlet characters $\mathrm{mod}q$, for every odd prime q up to 10⁷. The programs used and the results here described are collected at the following address http://www.math.unipd.it/~languasc/Littlewood_ineq.html.

让 $\mathrm{大号}（s，\chi ）$是与定义的非主要本原Dirichlet字符χ相关的Dirichlet L-函数$\mathrm{模}q$，其中q是奇数素数。在本文中，我们介绍了一种快速的计算方法$|\mathrm{大号}（\mathrm{1个}，\chi ）|$使用EulerΓ函数的值。我们还介绍了另一种计算方式$\mathrm{日志}\mathrm{\Gamma }（X）$ 和 $\psi （X）={\mathrm{\Gamma }}^{\prime }/\mathrm{\Gamma }（X）$， $X\in （0，\mathrm{1个}）$。使用这样的算法，我们在数值上验证了经典的Littlewood边界和最近的Lamzouri-Li-Soundararajan估计$|\mathrm{大号}（\mathrm{1个}，\chi ）|$，其中χ遍历非主要原始Dirichlet字符$\mathrm{模}q$，对于每个奇数质数q最多10 ⁷。在以下地址http://www.math.unipd.it/~languasc/Littlewood_ineq.html上收集了所使用的程序和此处描述的结果。