Further explorations of Boyd's conjectures and a conductor 21 elliptic curve

We prove that the (logarithmic) Mahler measure m(P ) of P (x, y) = x + 1/x + y + 1/y + 3 is equal to the L-value 2L′(E, 0) attached to the elliptic curve E : P (x, y) = 0 of conductor 21. In order to do this we investigate the measure of a more general Laurent polynomial Pa,b,c(x, y) = a ( x + 1 x ) + b ( y + 1 y ) + c and show that the wanted quantity m(P ) is related to a “half-Mahler” measure of P̃ (x, y) = P7,1,3(x, y). In the finale we use the modular parametrization of the elliptic curve P̃ (x, y) = 0, again of conductor 21, due to Ramanujan and the Mellit–Brunault formula for the regulator of modular units.