Disorder effects are especially pronounced around nodal points in linearly dispersing band structures as present in graphene or Weyl semimetals. Despite the enormous experimental and numerical progress, even a simple quantity like the average density of states cannot be assessed quantitatively by analytical means. We demonstrate how this important problem can be solved employing the functional renormalization group method, and, for the two-dimensional case, we demonstrate excellent agreement with reference data from numerical simulations based on tight-binding models. In three dimensions our analytic results also improve drastically on existing approaches.