Repeated integration by parts gives u k - 2 u / log(1 - t) log (-) [ t I ---------------------- dt ] t / 0 li (u) = - ----------------------------. k (k - 2)! Recklessly differentiating dk produces an integral that Mma expands into the termwise differentiated series (i.e., with a factor of -log n in the nth term). This series gives you (http://www.tweedledum.com/rwg/filds.dvi) t inf / / 2 2 [ [ s log(t + s ) I log(Gamma(x)) dx = I -------------- ds + t log(Gamma(t)) ] ] 2 %pi s / / %e - 1 1 0 2 2 t log(t) - t t 5 log(2 %pi) - %gamma + 1 zeta'(2) - ------------- + -- - ------------------------- - --------, 2 4 12 2 2 %pi which you can almost get from integrating Binet's second expansion of log Gamma(z) dz, but instead get mysterious integrals for some of the constants. Binet may have had this, but had no notation (nor popularity) for zeta'. For valuations of ilg(n/4 and n/6) (and hence the integral on the right), see www.tweedledum.com/rwg/idents.htm, (d1021) et seq. (near the end). The rhs integral (and Binet's) look suspiciously Abel-Plana, and might yield interesting series (or products!). With exp(ilg) (= "prodigal"(Gamma)), you can interpolate 1^1 2^2 ... n^n and 1! 2! ... n!, and derive "superStirling's" formulae. The reflection formula gives Catalan's constant. --rwg