Close, but wrong! Out[557]= -317-224 Sqrt[2]+120 Sqrt[7]+85 Sqrt[14]-2 Sqrt[100216+70864 Sqrt[2]-37878 Sqrt[7]-26784 Sqrt[14]] Is good to 1432 places: In[558]:= elliptic\[Alpha][14]=%; In[562]:= (1/\[Pi]-4\[Sqrt]r q D[Log@EllipticTheta [4,0,q],q])/EllipticTheta[3,0,q]^4/.q->E^(-\[Pi]\[Sqrt]r)/.r->14 Out[562]= (1/\[Pi]-(4 Sqrt[14] E^(-Sqrt[14] \[Pi]) (EllipticTheta^(0,0,1))[4,0,E^(-Sqrt[14] \[Pi])])/EllipticTheta[4,0,E^(-Sqrt[14] \[Pi])])/EllipticTheta[3,0,E^(-Sqrt[14] \[Pi])]^4 IIn[564]:= $MaxExtraPrecision = 999; N[-317 - 224 Sqrt[2] + 120 Sqrt[7] + 85 Sqrt[14] - 2 Sqrt[100216 + 70864 Sqrt[2] - 37878 Sqrt[7] - 26784 Sqrt[14]] - %562, 999] // tim During evaluation of In[564]:= N::meprec: Internal precision limit $MaxExtraPrecision = 999.` reached while evaluating -317-224 Sqrt[2]+120 Sqrt[7]+85 Sqrt[14]-2 Sqrt[100216+70864 Sqrt[2]-37878 Sqrt[7]-26784 Sqrt[14]]-(1/\[Pi]-(4 Sqrt[14] E^<<1>> (EllipticTheta^(0,0,1))[4,0,E^<<1>>])/EllipticTheta[4,0,E^Times[<<1>>]])/EllipticTheta[3,0,E^(-Sqrt[14] \[Pi])]^4. During evaluation of In[564]:= 35.165791,0 Out[564]= \ -2.1780143214865091427625118320922249681292841601643494815193895973682\ 8186997202519647922211881399689317439744967456115770875576389279894343\ 4103510213787195402499227137612713482842417669344355938557334053047424\ 3665569026553937773794102097792619181414582388205436930400880121908180\ 2624782639365438650454014709016211078611712300754946583633339665388728\ 7464787205668010923217022021776141717363845190854652724894091969028686\ 4133107617200474961151654220332919509637503393308771565207572946486876\ 8564580732589358492023185254688451674663015301074951385960971393120855\ 8975*10^-1433 —rwg On Wed, Nov 14, 2018 at 7:15 PM Bill Gosper <billgosper@gmail.com> wrote:
Numerically, I get -317 - 224 Sqrt[2] + 120 Sqrt[7] + 85 Sqrt[14] - 2 Sqrt[100216 + 70864 Sqrt[2] - 37878 Sqrt[7] - 26784 Sqrt[14]] —rwg