Linear Mahler measures and double \(L\)-values of modular forms (Q403288)

The equality NEWLINE\[NEWLINEm(P)={1\over2\pi i)^n}\int_{|z_1|=\cdots=|z_n|=1}{\log|P(x_1,\dots,x_n)|\over x_1\cdots x_n}dx_1\cdots x_nNEWLINE\]NEWLINE defines the logarithm of Mahler measure of a polynomial \(P(x_1,\dots,x_n)\). It has been shown by \textit{C. J. Smyth} [Bull. Aust. Math. Soc. 23, 49--63 (1981; Zbl 0442.10034)] that NEWLINE\[NEWLINEm(1+x_1+x_2)={3\sqrt3\over4\pi}L(\chi_{3},2),\tag{1}NEWLINE\]NEWLINE and in Appendix 1 of \textit{D. W. Boyd}'s paper [Can. Math. Bull. 24, 453--469 (1981; Zbl 0474.12005)]NEWLINE\[NEWLINEm(1+x_1+x_2+x_3) = {7\over2\pi^2}\zeta(3).\tag{2}NEWLINE\]NEWLINENEWLINENEWLINENEWLINEThe author obtains a rather complicated formula expressing \(m(1+x_1+x_2+x_3+x_4)\) by integrals of modular forms.