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Setting $E_{k}(z)=\frac{G_k(z)}{\zeta(k)}$ for $G_k$ defined as in the question, it is well known that $$E_{4k}(i)=\frac{1}{2\zeta(4k)}\left(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx\right)^{4k}\frac{H_{4k}}{(4k)!}, $$ where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ are the first two such numbers, but in general these come as coefficients in the Laurent series of Weierstrass's elliptic function. The introduction of Tsumura's paper On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type gives more details, and the relevant paper by Hurwitz is cited there.

It is well known that $$G_{4k}(i)=\left(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx\right)^{4k}\frac{H_{4k}}{(4k)!}, $$ where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ are the first two such numbers, but in general these come as coefficients in the Laurent series of Weierstrass's elliptic function. The introduction of Tsumura's paper On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type gives more details, and the relevant paper by Hurwitz is cited there.

Setting $E_{k}(z)=\frac{G_k(z)}{\zeta(k)}$ for $G_k$ defined as in the question, it is well known that $$E_{4k}(i)=\frac{1}{2\zeta(4k)}\left(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx\right)^{4k}\frac{H_{4k}}{(4k)!}, $$ where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ are the first two such numbers. The introduction of Tsumura's paper On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type gives more details, and the relevant paper by Hurwitz is cited there.

Setting $E_{k}(z)=\frac{G_k(z)}{\zeta(k)}$ for $G_k$ defined as in the question, it is well known that $$E_{4k}(i)=\frac{1}{2\zeta(4k)}\left(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx\right)^{4k}\frac{H_{4k}}{(4k)!}, $$ where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ are the first two such numbers, but in general these come as coefficients in the Laurent series of Weierstrass's elliptic function. The introduction of Tsumura's paper On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type gives more details, and the relevant paper by Hurwitz is cited there.

It is well known that $G_{4k}(i)=(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx)^{4k}\frac{1}{(4k)!} H_{4k}$, where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ are the first two such numbers. The introduction of Tsumura's paper On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type gives more details, and the relevant paper by Hurwitz is cited there.

Setting $E_{k}(z)=\frac{G_k(z)}{\zeta(k)}$ for $G_k$ defined as in the question, it is well known that $$E_{4k}(i)=\frac{1}{2\zeta(4k)}\left(4\int_{0}^1\frac{1}{\sqrt{1-x^4}}dx\right)^{4k}\frac{H_{4k}}{(4k)!}, $$ where $H_{4k}$ are called Hurwitz numbers. $H_4=\frac{1}{10}$ and $H_{8}=\frac{3}{10}$ are the first two such numbers. The introduction of Tsumura's paper On certain analogues of Eisenstein series and their evaluation formulas of Hurwitz type gives more details, and the relevant paper by Hurwitz is cited there.