The eighth singular value: http://www.wolframalpha.com/input/?i=EllipticK[-112+-+80+Sqrt[2]+%2B+4+Sqrt[2+%28799+%2B+565+Sqrt[2]%29]]%2F++EllipticK[113+%2B+80+Sqrt[2]+-+4+Sqrt[2+%28799+%2B+565+Sqrt[2]%29]]+%3D%3D+++2+Sqrt[2] http://www.wolframalpha.com/input/?_=1353320405208&i=EllipticK[113+%2b+80+Sqrt[2]+-+4+Sqrt[2+%28799+%2b+565+Sqrt[2]%29]]+%3d%3d+++1%2f16+%281%2f2+%281+%2b+Sqrt[2]%29%29 ^%281%2f4%29+Sqrt[%28+++2+Sqrt[2]+%2b+Sqrt[1+%2b+5+Sqrt[2]]%29%2f\[Pi]]+Gamma[1%2f8]+Gamma[3%2f8]&fp=1&incTime=true If this times out, try http://www.wolframalpha.com/input/?i={EllipticK[113+%2B+80+Sqrt[2]+-+4+Sqrt[2+%28799+%2B+565+Sqrt[2]%29]]+%2C++1%2F16+%281%2F2+%281+%2B+Sqrt[2]%29%29 ^%281%2F4%29+Sqrt[%28+++2+Sqrt[2]+%2B+Sqrt[1+%2B+5+Sqrt[2]]%29%2F\[Pi]]+Gamma[1%2F8]+Gamma[3%2F8]} Found by accident: http://www.wolframalpha.com/input/?i=EllipticK[-112+%2B+80+Sqrt[2]+%2B+4+I+Sqrt[2+%28-799+%2B+565+Sqrt[2]%29]]%2F++EllipticK[113+-+80+Sqrt[2]+-+4+I+Sqrt[2+%28-799+%2B+565+Sqrt[2]%29]]+%3D%3D+%28+++2+I%29%2F3+%2B+%282+Sqrt[2]%29%2F3 http://www.wolframalpha.com/input/?i={EllipticK[113-80Sqrt[2]-4I+Sqrt[2%28-799%2B565Sqrt[2]%29]]%2C%28Sqrt[3%281%2BSqrt[2]%29]E ^%28-I+ArcTan[1%2F23Sqrt[2]%287%2B6Sqrt[2]-3Sqrt[-7%2B17Sqrt[2]]%29]%29Gamma[1%2F8]Gamma[3%2F8]%29%2F%2816+2^%281%2F4%29Sqrt[%CF%80]%29} To view two side by side, I was able to open a second Alpha in a separate Firefox window. Beware: GMail may have inserted linebreaks. --rwg Also, Weisstein's Mathworld has more K special values than Borwein&Borwein. Also, that Byrd&Friedman Handbook of Elliptic Integrals appears to have only the small modulus Fourier series for am, whereas the pendulum formula requires http://www.wolframalpha.com/input/?i={JacobiAmplitude[u%2Cm]%2C+Sum[%28Sech[%CF%80*Sqrt[m]*%28n-1%2F2%29*EllipticK[1-m]%2FEllipticK[1%2Fm]]*Sin[%28%CF%80*Sqrt[m]*%28n-1%2F2%29*u%29%2FEllipticK[1%2Fm]]%29%2F%28n-1%2F2%29%2C{n%2C1%2CHoldForm[%E2%88%9E]}]} with briskly exponential convergence. So I still lack evidence that this useful formula is actually in use. (The HoldForm is to prevent Alpha timeout.) --rwg