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 $d$.
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$\begingroup$ Also, if someone can kindly point me to instructions for including an image with a question, I would post a cute graph here. $\endgroup$– Michael BeesonCommented Nov 18, 2022 at 16:47

6$\begingroup$ 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 $2d$ makes that factor $1$, which we can expect to roughly halve the maximal $L(1,\chi_d)$. $\endgroup$– Noam D. ElkiesCommented Nov 18, 2022 at 17:00

$\begingroup$ For images, see the last paragraphs at mathoverflow.net/help/formatting $\endgroup$– Gerry MyersonCommented Nov 18, 2022 at 22:07
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