The generalized Quillen–Lichtenbaum conjecture for Artin L-functions

Let $F'/F$ be a $G$-Galois extension of number fields, where $G$ is finite, and let $\rho\colon G\to \mathrm{GL}_N(\mathcal{O}_E)$ be an $E$-linear Galois representation for a number field $E$. Let $L^*(\mathcal{O}_F,\rho,1-k)$ denote the leading coefficient of the Taylor series of the associated $L$-function at $s=1-k$, and let $R_{k,\rho}$ be the $k$-th regulator of $\rho$. Generalized Quillen–Lichtenbaum conjecture. Up to a power of $2$, for $k\geq 1$ one has

$$\prod_{\tau\in \operatorname{Gal}(E/\mathbb{Q})} L^*(\mathcal{O}_F,\tau\circ \rho,1-k)=\pm \frac{\#\pi^G_{2k-2}(\mathrm{K}(\mathcal{O}_{F'})\otimes \mathrm{M}(\rho))}{\#\pi^G_{2k-1}(\mathrm{K}(\mathcal{O}_{F'})\otimes \mathrm{M}(\rho))_{\mathrm{torsion}}}\cdot \prod_{\tau\in \operatorname{Gal}(E/\mathbb{Q})} R_{k,\tau\circ \rho}.$$

This generalizes the Quillen–Lichtenbaum relation from Dedekind zeta values to Artin $L$-functions. The paper proves the formula in many cases, while the stated assumptions leave the general number-field and representation-theoretic cases open.