Question Number 221981 by MrGaster last updated on 14/Jun/25
$$\mathrm{Prove}:\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{{n}^{\mathrm{3}} }{{e}^{\mathrm{2}\pi{n}} −\mathrm{1}}=\frac{\Gamma\left(\frac{\mathrm{1}}{\mathrm{4}}\right)^{\mathrm{8}} }{\mathrm{5120}\pi^{\mathrm{6}} }−\frac{\mathrm{1}}{\mathrm{240}}=\frac{\mathrm{1}}{\mathrm{80}}\left(\frac{\varpi}{\pi}\right)^{\mathrm{4}} −\frac{\mathrm{1}}{\mathrm{240}} \\ $$