Denoted zn(u, k) or Z(u). Z(ϕ| m) congruent E(ϕ| m) - (E(m) F(ϕ| m))/(K(m)), where ϕ is the Jacobi amplitude, m is the parameter, and F(ϕ| m) and K(m) are elliptic integrals of the first kind, and E(m) is an elliptic integral of the second kind. See Gradshteyn and Ryzhik (2000, p. xxxi) for expressions in terms of theta functions. The Jacobi zeta functions is implemented in the Wolfram Language as JacobiZeta[phi, m].

elliptic integral of the first kind | elliptic integral of the second kind | Heuman lambda function | zeta function

JacobiZeta

Carl Jacobi