Here's a particularly simple one, given Gamma(1/4): Sqrt[Beta[1/8,1/8]]== 4*EllipticTheta[3, 0, 1/(E^(4*Sqrt[2]*Pi))]* Sqrt[Pi]/((2 - Sqrt[2])^(1/4) + 1/2^(1/8)) Not spectacular, but it's so simple, why not take three terms of theta[3]? {Sqrt[Beta[1/8, 1/8]], 4*(2/E^(16*Sqrt[2]*Pi) + 2/E^(4*Sqrt[2]*Pi) + 1)*Sqrt[Pi]/((2 - Sqrt[2])^(1/4) + 1/2^(1/8))} N[%, 72] {3.95668714346392620150868646519961135227029008333530237247887292531488627, 3.95668714346392620150868646519961135227029008333530237247887292531488354} --rwg