Dirichlet's $L$-function plays a central role in analytic number theory. For any integer $d\equiv0,1\pmod4$, let $$L_d(2):=L\left(2,\left(\frac{d}{\cdot}\right)\right)=\sum_{k=1}^\infty\frac{(\frac dk)}{k^2},$$ where $(\frac d{\cdot})$ denotes the Kronecker symbol. How to compute the value of $L_d(2)$ efficiently? For this purpose, we should seek for fast converging series related to $L_d(2)$.
The number $G=L_{-4}(2)$ is called the Catalan constant. For discussions on fast converging series for $G$, one may consult Question 424055 at MathOverflow.
The constant $K=L_{-3}(2)$ was proved to be irrational in arXiv:2408.15403 by F. Calegari, V. Dimitrov and Y. Tang. Recently, I conjectured that $$\begin{aligned}&\sum_{k=1}^\infty\frac{(54(265-153\sqrt3))^k}{k^3\binom{2k}k^2\binom{3k}k} \left(3k(17\sqrt3+27)-16\sqrt3-27\right) \\&\qquad =135\left(G-\frac{11}{16}\sqrt3 K\right). \end{aligned}\tag{1}$$ Note that the series in $(1)$ has converging rate about $-1/530$.
For $L_{-7}(2)$, in arXiv:2506.01865 I and Yajun Zhou established the identity $$\sum_{k=1}^\infty\frac{(3\sqrt7-8)^{3k}}{k^3\binom{2k}k^3}(3k(85\sqrt7+224)-8(14\sqrt7+37))=32G-\frac{77}8\sqrt7L_{-7}(2).\tag{2}$$ The converging rate of the series in $(2)$ is about $-1/259072$.
For $L_{-8}(2)$, in arXiv:2506.01865 I and Yajun Zhou established the identity $$\begin{aligned}&\sum_{k=1}^\infty\frac{(-1)^k(2-\sqrt2)^{12k}}{k^3\binom{2k}k^3}(132k(5\sqrt2+7)-284\sqrt2-401) \\&\qquad =16(23G-14\sqrt2L_{-8}(2)).\end{aligned}\tag{3}$$ The converging rate of the series in $(3)$ is about $-1/39202$.
For $L_{-11}(2)$, in arXiv:2408.15403 I conjectured that $$\begin{aligned}&\sum_{k=1}^\infty\frac{(32(91\sqrt{33}-523))^{k}}{k^3\binom{2k}k^2\binom{3k}k} \left((91\sqrt{33}+891)k-33\sqrt{33}-225\right) \\&\qquad=320\left(\frac{11}3\sqrt{33}L_{-11}(2)-27K\right).\end{aligned}\tag{4}$$ The series in $(4)$ has converging rate about $-1/13.7868$
For $L_{-20}(2)$ and $L_{-24}(2)$, Guillera and Rogers [J. Aust. Math. Soc. 97 (2014), 78-106] obtained the identities $$\sum_{k=1}^\infty\frac{(-64)^k(\sqrt5-2)^{4k}}{k^3\binom{2k}k^3}(12k(2\sqrt5+5)-10\sqrt5-23)=40\sqrt5L_{-20}(2)-112G\tag{5}$$ and $$\begin{aligned}&\sum_{k=1}^\infty\frac{(-64)^k(5-2\sqrt6)^{4k}}{k^3\binom{2k}k^3}(28k(2\sqrt6+5)-24\sqrt6-59) \\&\qquad =32\sqrt6L_{-24}(2)-\frac{272}3G.\end{aligned}\tag{6}$$ The series in $(5)$ and $(6)$ have converging rates about $-1/322$ and $-1/9602$ respectively.
For $L_{-39}(2)$, in arXiv:2408.15403 I conjectured that $$\begin{aligned}&\sum_{k=1}^\infty\frac{(-1728)^k(18-5\sqrt{13})^{2k}}{k^3\binom{2k}k^2\binom{3k}k} (15k(9\sqrt{13}+26)-39\sqrt{13}-134) \\&\qquad=54(80K-13\sqrt{13}L_{-39}(2)).\end{aligned}\tag{7}$$ The series in $(7)$ has converging rate about $-1/81.125$
For $L_{-56}(2)$, in arXiv:2506.01865 I and Yajun Zhou established the identity $$\begin{aligned}&\sum_{k=1}^\infty\frac{(-2^{11})^k(45-17\sqrt7)^{2k}}{k^3\binom{2k}k^3}(6k(17\sqrt7+35)-35\sqrt7-89) \\&\qquad=128(20L_{-8}(2)-7\sqrt7L_{-56}(2)). \end{aligned}\tag{8}$$ The converging rate of the series in $(8)$ is about $-1/63.25$.
For $L_{-116}(2)$, in arXiv:2506.01865 I and Yajun Zhou established the identity $$\begin{aligned}&\sum_{k=1}^\infty\frac{(-1)^k(5\sqrt{29}-27)^{6k}}{k^3\binom{2k}k^3}(132k(70\sqrt{29}+377)-4234\sqrt{29}-22801) \\&\qquad=\frac 83(29\sqrt{29}L_{-116}(2)-198G). \end{aligned}\tag{9}$$ The converging rate of the series in $(9)$ is about $-1/384238402$.
I also have conjectured some fast converging series for computing $L_{-68}(2), L_{-87}(2)$ and $L_{-111}(2)$. For details, one may consult Section 3 of arXiv:2408.15403.
Questions. For $d=-3,-7,-8,-11,-20,-24,-39,-56,-116$, whether $(1)-(9)$ provide the fastest way to compute $L_d(2)$ respectively? Can one find more negative integers $d\equiv0,1\pmod 4$ such that there is an essentially new series with a geometric rate for computing $L_{d}(2)$?
Your comments are welcome!
