The Group is the Subgroup of the Gamma Group with and
Odd; and Even. The function

(1) |

(2) |

gives the value of the Modulus
for which the complementary and normal complete Elliptic Integrals of
the First Kind are related by

(3) |

(4) |

(5) |

From the definition of the lambda function,

(6) |

(7) | |||

(8) |

Special values are

In addition,

**References**

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.

Selberg, A. and Chowla, S. ``On Epstein's Zeta-Function.'' *J. Reine. Angew. Math.* **227**, 86-110, 1967.

Watson, G. N. ``Some Singular Moduli (1).'' *Quart. J. Math.* **3**, 81-98, 1932.

© 1996-9

1999-05-25