  Given a Coxeter system $(W,S)$ and a multiparameter $\mathbf{q}$ of real
numbers indexed by $S$, one can define the weighted $L^2$-cohomology groups and
associate to them a nonnegative real number called the weighted $L^2$-Betti
number. We show that for ranges of $\mathbf{q}$ depending on certain subgroups
of $W$, the weighted $L^2$-cohomology groups of $W$ are concentrated in low
dimensions. We then prove new vanishing results for the weighted
$L^2$-cohomology of certain low-dimensional Coxeter groups. Our arguments rely
on computing the $L^2$-cohomology of certain complexes called ruins, as well as
the resolution of the Strong Atiyah Conjecture for hyperbolic Coxeter groups.
We conclude by extending to the weighted setting the computations of Davis and
Okun for the case where the nerve of a right-angled Coxeter group is the
barycentric subdivision of a PL-cellulation of an $(n-1)$-manifold with
$n=6,8$.
