  In this paper we prove the Leibniz analogue of Whitehead's vanishing theorem
for the Chevalley-Eilenberg cohomology of Lie algebras. As a consequence, we
obtain the second Whitehead lemma for Leibniz algebras. Moreover, we compute
the cohomology of several Leibniz algebras with adjoint or irreducible
coefficients. Our main tool is a Leibniz analogue of the Hochschild-Serre
spectral sequence, which is an extension of (the dual of) a spectral sequence
of Pirashvili for Leibniz homology from symmetric bimodules to arbitrary
bimodules.
