Let L/K be a finite Galois extension of number fields with Galois group G . Let p be an odd prime and r>1 be an integer. Assuming a conjecture of Schneider, we formulate a conjecture that relates special values of equivariant Artin L -series at s=r to the compact support cohomology of the étale p -adic sheaf \mathbb{Z}_p(r) . We show that our conjecture is essentially equivalent to the p -part of the equivariant Tamagawa number conjecture for the pair (h^0(\text{Spec}(L))(r),\mathbb{Z}[G]) . We derive from this explicit constraints on the Galois module structure of Banaszak's p -adic wild kernels.