Question

Prove $\int_0^\infty\Big|\Gamma\Big(\frac13+\frac{i\alpha}{\pi}\Big)\Big|^6\alpha \sinh\alpha~d\alpha=\frac{\pi\sqrt{27}}{4}\Gamma^9\Big(\frac23\Big)$

score 501 · Answer 1 · 2026-04-16 14:25:45

501

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Prove $\int_0^\infty\Big|\Gamma\Big(\frac13+\frac{i\alpha}{\pi}\Big)\Big|^6\alpha \sinh\alpha~d\alpha=\frac{\pi\sqrt{27}}{4}\Gamma^9\Big(\frac23\Big)$

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52

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List Question

Prove $\int_0^\infty\Big|\Gamma\Big(\frac13+\frac{i\alpha}{\pi}\Big)\Big|^6\alpha \sinh\alpha~d\alpha=\frac{\pi\sqrt{27}}{4}\Gamma^9\Big(\frac23\Big)$

Finding $x$ where inequality is satisfied for sure

Bessel equation of the first order has the solution $y=xJ_1(x)$.

Can anyone explain the inverse regularized beta function?

Factorial related sum

What functions is this?

Calculate a $\operatorname{Li}_{2}(-1)$ using Integral Representation

Integral representation of Legendre function of the second kind

Integral $\int_{-1}^{1} \frac{1}{(1+u^2)^{n/2}} \exp{\left(-2\pi \frac{a^2+b^2}{1+u^2}\right)} \exp{\left(-4\pi i ab \frac{u}{1+u^2}\right)} du $

Beta function inequality

Analytic continuation of an integral involving the derivative of the Riemann zeta function

From Sophie Germain's Identity to difficult integrals: the integral $\int_0^1\int_0^1\cos\left((x+y)^2+y^2\right)dxdy$

Accurate numerical evaluation of a ratio of Bessel functions

Function with extrema 1, 1/4, 1/9, ...

A triple integral involving $\text{Li}_2$

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