The weak Chinburg conjecture on Mahler measures

For every negative fundamental discriminant $-f$ and every $k\geq1$, we construct a rational function $R_{f,2k}\in\mathbb{Q}(x_1,\ldots,x_{2k})$ and a constant $r_{f,2k}\in\mathbb{Q}^\times$ such that \[ m(R_{f,2k})=r_{f,2k}L'(χ_{-f},1-2k), \] where $m$ denotes the logarithmic Mahler measure and $χ_