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We prove a formula relating the Mahler measure of an infinite family of three-variable polynomials to a combination of the Riemann zeta function at s = 3 and special values of the Bloch–Wigner dilogarithm by evaluating a regulator. The evaluation requires two different applications of Jensen's formula and analyzing the integral in two different planes, as opposed to the more common strategy of using only one plane. The degrees of the monomials involving one of the variables are allowed to vary freely, leading to an interesting application of the Boyd–Lawton formula. [ABSTRACT FROM AUTHOR]