# $L(1,\chi_d)$ for even d

This question concerns the asymptotic maximum of $$L(1,\chi_d)/ \log \log d$$ when $$d$$ is even. I computed it for even $$d$$ less than 7.5 million. The largest value is $$1.230126$$, occurring for $$d= 2552806$$. This value is never exceeded for the next five million $$d$$. Languasco (arXiv:2005.04664v2 [math.NT] 26 May 2020) made similar numerical computations without assuming $$d$$ is even and got a much larger maximum of 2.21. Can anyone explain why $$L(1,\chi_d)$$ should be smaller for even $$d$$? In particular, for infinitely many $$d$$ we know (on GRH maybe) that the value $$e^{\gamma} = 1.781\ldots$$ is exceeded, but apparently not by even $$d$$.

• Also, if someone can kindly point me to instructions for including an image with a question, I would post a cute graph here. Commented Nov 18, 2022 at 16:47
• One heuristic model for the distribution of $L(1,\chi_d)$ is a product of Euler factors $(1 - \chi(p)/p)^{-1}$ where each $\chi(p)$ is chosen independently from $\{-1,0,+1\}$ with appropriate probabilities. The $p=2$ factor has the largest effect, and can be $2$, $1$, or $2/3$. Requiring $2|d$ makes that factor $1$, which we can expect to roughly halve the maximal $L(1,\chi_d)$. Commented Nov 18, 2022 at 17:00
• For images, see the last paragraphs at mathoverflow.net/help/formatting Commented Nov 18, 2022 at 22:07