Singular values of generalized $\lambda$ functions

Abstract

We study special values of a modular function $\Lambda$ which is one of generalized $\lambda$ functions. We show special values of $\Lambda$ at imaginary quadratic points are algebraic integers. Further we prove that $\Lambda$ and the modular invariant function generate the modular function field with respect to the modular subgroup ${\Gamma }_1(N)$.