* Bill Gosper <billgosper@gmail.com> [Feb 13. 2011 17:44]:
For the 17th singular value, Mathworld gives a nonsensical surdoma with a chunky subexpression both added and subtracted. I PSLQed for it a big bioctic and manually massaged it down to LambdaStar[17]==1/8 (3 + Sqrt[17]) (2 Sqrt[2] - Sqrt[1 + Sqrt[17]]) - Sqrt[-19 Sqrt[2] - 5 Sqrt[34] + Sqrt[1 + Sqrt[17]] (13 + 3 Sqrt[17])]/(2 2^(1/4)) which the denester considers finished.
Your's (with those precious '*'s): t=1/8*(3+sqrt(17))*(2*sqrt(2)-sqrt(1+sqrt(17)))-sqrt(-19*sqrt(2)-5*sqrt(34)+sqrt(1+sqrt(17))*(13+3*sqrt(17)))/(2*2^(1/4)) /* == 0.006156331061153274511510448... */ /* == Sage gives, by solving the minimal polynomial: */ sqrt(-1/2*sqrt(20*sqrt(1649*sqrt(17)+6799)*sqrt(2)-400*sqrt(17)-1648)+1/2) Dunno whether this is a match in any regard
--rwg [...]
Let's not forget Broadhurst's very neat expressions: Message-ID: <200803140250.m2E2oHi4002315@philter.princeton.idaccr.org> Date: Thu, 13 Mar 2008 22:50:17 -0400 From: David Broadhurst <D.Broadhurst@open.ac.uk> To: NMBRTHRY@LISTSERV.NODAK.EDU Reply-To: David Broadhurst <D.Broadhurst@open.ac.uk> Subject: Singular value for Euler's numerus idoneus N=1365 default(realprecision,1050); a_1 = 550*sqrt(13) + 318*sqrt(39) + 750*sqrt(7) + 433*sqrt(21); a_2 = 1986 + 1145*sqrt(3) + 208*sqrt(91) + 120*sqrt(273); b = ((a_1-a_2)^2 - 2)/2; k_1365 = sqrt(1/2 - b^2/4*sqrt(4-b^2) - (2-b^2)/4*sqrt(1-b^2)); c = (33*sqrt(13) + 3*sqrt(39) - 33*sqrt(7) - 11*sqrt(21))/24/b; d =(sqrt(4-b^2) + sqrt(1-b^2))/sqrt(2); G = prod(k=1,5460,if(gcd(5460,k)==1,gamma(k/5460)^kronecker(-5460,k),1)); K_1365 = sqrt(Pi*c*d/1365)*G^(1/32); print(agm(1,sqrt(1-k_1365^2))/agm(1,k_1365) - sqrt(1365)); print(K_1365 - Pi/2/agm(1,sqrt(1-k_1365^2)));