Remark on upper bounds for \(L(1, \chi)\) (Q1924891)

Let \(p\) be an odd prime with \(p\equiv 1\pmod 4\). Further, let \(H(p)=\sqrt p(\log\sqrt p+1)/2\), \(W(p)=\sqrt p(\log\sqrt p+\gamma-1/2)/2\) and \(O(p)=\log(2(\begin{smallmatrix} n+\omega\\ n\end{smallmatrix})/\sqrt p)\), where \(n=(p-1)/4\), \(\omega=(1+\sqrt p)/2\) and \(\nu\) is Euler's constant. In this paper, the author proves that \[ \begin{aligned} O(p) &<W(p)< H(p)\qquad \text{if }5\leq p \leq 661\\ W(p) &<O(p)< H(p)\qquad \text{otherwise}.\end{aligned} \] The above inequalities are concerned with upper bounds for the value \(L(1,\chi)\) of Dirichlet \(L\)-functions.