Three-variable Mahler measures and special values of modular and Dirichlet $L$-series

Abstract

In this paper we prove that the Mahler measures of the Laurent polynomials $(x+x^{-1})(y+y^{-1})(z+z^{-1})+k$, $(x+x^{-1}{)}^2(y+y^{-1}{)}^2(1+z{)}^3{z}^{-2}-k$, and $x^4+y^4+z^4+1+k^{1/4}xyz$, for various values of $k$, are of the form $r_1{L}^{\prime }(f,0)+r_2{L}^{\prime }(\chi ,-1)$, where $r_1,r_2\in \mathbb{Q}$, $f$ is a CM newform of weight 3, and $\chi$ is a quadratic character. Since it has been proved that these Maher measures can also be expressed in terms of logarithms and ${}_5{F}_4$-hypergeometric series, we obtain several new hypergeometric evaluations and transformations from these results.

Cited by in corpus (2)

• Mahler measures as linear combinations of $L$-values of multiple modular forms
• On a non-critical symmetric square $L$-value of the congruent number elliptic curves