Notice that the above results are considerably simpler than the 5th singular values tabulated in MathWorld and Borwein&Borwein (p 298), due partly to using the 20th singular value instead, but more to a Gamma simplification that was not in http://arxiv.org/abs/math/0403510 .
Actually, it sort of was, giving Gamma of n/20 in terms of 1/20, 1/5, and 2/5. But that's three species vs two. They also give n/24 in terms of 1/24, 1/3, and 1/4, which with
rwg>This pushes the luck frontier back to, e.g., e^-(pi sqrt(3/2)).
eta(%e^-(2*sqrt(2)*%pi/sqrt(3)))=Gamma(1/24)*sqrt(sin(%pi/24))*csc(%pi/8)^(1/6) /(2^(23/24)*3^(1/8)*(sqrt(3)-1)^(1/4)*sqrt(%pi)*sqrt(Gamma(1/12)))
2 sqrt(2) %pi - ------------- sqrt(3) eta(%e ) = 1 %pi 1/6 %pi Gamma(--) sqrt(sin(---)) csc (---) 24 24 8 ------------------------------------------------------, 23/24 1/8 1/4 1 2 3 (sqrt(3) - 1) sqrt(%pi) sqrt(Gamma(--)) 12 gives us AGMs for Gamma(n/24). The table also indicates that eta(e^-(pi sqrt 15)) needs only Gamma of 1/15 an 4/15. Rashly assuming I am the only current source for these results, if anyone wants a particular eta, eta', or logderiv series(e^(-pi sqrt(rational)), send it along. (Meanwhile, I'll resume jobseeking.) Finally, those log derivative sums came in pairs by virtue of the imaginary transformation: x*('sum(n/(%e^(2*%pi*n*x)-1),n,1,inf)-1/24)+('sum(n/(%e^(2*%pi*n/x)-1),n,1,inf)-1/24)/x = -1/(4*%pi) inf inf ==== ==== \ n 1 1 \ n 1 1 x ( > --------------- - --) + - ( > ------------- - --) = - -----. / 2 %pi n x 24 x / 2 %pi n 24 4 %pi ==== %e - 1 ==== ------- n = 1 n = 1 x %e - 1 --rwg