The elliptic lambda function is essentially the same as the inverse nome, the difference being that elliptic lambda function is a function of the
half-period ratio , while the inverse nome
is a function of the nome , where is itself a function of .

For all rational , and are known as elliptic
integral singular values, and can be expressed in terms of a finite number of
gamma functions (Selberg and Chowla 1967). Values
of
for small
include

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

(27)

where

(28)

The algebraic orders of these are given by 2, 2, 4, 2, 8, 4, 4, 4, 8, 4, 12, 4, 8,
8, 8, 4, ... (OEIS A084540).

Some additional exact values are given by

(29)

(30)

(31)

(32)

(33)

(34)

Exact values can also be found for rational , including

Borwein, J. M. and Borwein, P. B. Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity.
New York: Wiley, pp. 139 and 298, 1987.Bowman, F. Introduction
to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95,
and 98, 1961.Conway, J. H. and Norton, S. P. "Monstrous
Moonshine." Bull. London Math. Soc.11, 308-339, 1979.Selberg,
A. and Chowla, S. "On Epstein's Zeta-Function." J. reine angew. Math.227,
86-110, 1967.Sloane, N. J. A. Sequences A029845,
A084540, and A115977
in "The On-Line Encyclopedia of Integer Sequences."Watson,
G. N. "Some Singular Moduli (1)." Quart. J. Math.3,
81-98, 1932.