Linear Mahler measures and double L-values of modular forms

We consider the Mahler measure of the polynomial 1+x1+x2+x3+x41+x1+x2+x3+x4, which is the first case not yet evaluated explicitly. A conjecture due to F. Rodriguez-Villegas represents this Mahler measure as a special value at the point 4 of the L-function of a modular form of weight 3. We prove that this Mahler measure is equal to a linear combination of double L-values of certain meromorphic modular forms of weight 4.