The congruent number elliptic curves are defined by $E_{d}:y^{2}=x^{3}-d^{2}x$, where $d\in \mathbb{N}$. We give a simple proof of a formula for $L(\operatorname{Sym}^{2}(E_{d}),3)$ in terms of the determinant of the elliptic trilogarithm evaluated at some degree zero divisors supported on the torsion points on $E_{d}(\overline{\mathbb{Q}})$.

We study the Mahler measures of certain families of Laurent polynomials in two and three variables. Each of the known Mahler measure formulas for these families involves $L$–values of at most one newform and/or at most one quadratic character. In this paper we show, either rigorously or numerically, that the Mahler measures of some polynomials are related to $L$–values of multiple newforms and quadratic characters simultaneously. The results suggest that the number of modular $L$–values appearing in the formulas significantly depends on the shape of the algebraic value of the parameter chosen for each polynomial. As a consequence, we also obtain new formulas relating special values of hypergeometric series evaluated at algebraic numbers to special values of $L$–functions.

We present another example of a 3-variable polynomial defining a $K3$-hypersurface and having a logarithmic Mahler measure expressed in terms of a Dirichlet $L$-series.