On Eulerian log-gamma integrals and Tornheim–Witten zeta functions

Stimulated by earlier work by Moll and his coworkers (Amdeberhan et al., Proc. Am. Math. Soc., 139(2):535–545, 2010 ), we evaluate various basic log Gamma integrals in terms of partial derivatives of Tornheim–Witten zeta functions and their extensions arising from evaluations of Fourier series. In particular, we fully evaluate ℒ𝒢_n=∫_0^1 log^n(x) dx for 1≤ n ≤4 and make some comments regarding the general case. The subsidiary computational challenges are substantial, interesting and significant in their own right.