When the elliptic modulus has a singular value, the complete elliptic integrals may
be computed in analytic form in terms of gamma functions.
Abel (quoted in Whittaker and Watson 1990, p. 525) proved that whenever

Values of for small integer in terms of gamma functions
are summarized below.

(3)

(4)

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

where
is the gamma function and is an algebraic number (Borwein and Borwein 1987, p. 298).

Borwein and Zucker (1992) give amazing expressions for singular values of complete
elliptic integrals in terms of central beta functions

(21)

Furthermore, they show that is always expressible in terms of these functions
for .
In such cases, the functions appearing in the expression are of
the form where and . The terms in the numerator depend on the sign of the
Kronecker symbol . Values for the first few are

and
is an algebraic number (Borwein and Zucker 1992). Note that is the only value in the above list which cannot be
expressed in terms of central beta functions.

Abel, N. H. "Recherches sur les fonctions elliptiques." J. reine angew. Math.3, 160-190, 1828. Reprinted in Abel, N. H.
Oeuvres
Completes (Ed. L. Sylow and S. Lie). New York: Johnson Reprint
Corp., p. 377, 1988.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.Borwein, J. M. and
Zucker, I. J. "Elliptic Integral Evaluation of the Gamma Function at Rational
Values of Small Denominator." IMA J. Numerical Analysis12, 519-526,
1992.Bowman, F. Introduction
to Elliptic Functions, with Applications. New York: Dover, pp. 75, 95,
and 98, 1961.Glasser, M. L. and Wood, V. E. "A Closed
Form Evaluation of the Elliptic Integral." Math. Comput.22, 535-536,
1971.Selberg, A. and Chowla, S. "On Epstein's Zeta-Function."
J. reine angew. Math.227, 86-110, 1967.Whittaker, E. T.
and Watson, G. N. A
Course in Modern Analysis, 4th ed. Cambridge, England: Cambridge University
Press, pp. 524-528, 1990.Wrigge, S. "An Elliptic Integral
Identity." Math. Comput.27, 837-840, 1973.Zucker,
I. J. "The Evaluation in Terms of -Functions of the Periods of Elliptic Curves Admitting
Complex Multiplication." Math. Proc. Cambridge Philos. Soc.82,
111-118, 1977.Zucker, I. J. and Joyce, G. S. "Special
Values of the Hypergeometric Series II." Math. Proc. Cambridge Philos. Soc.131,
309-319, 2001.