  We prove a lower bound for the entropy dissipation of the Landau equation
with Coulomb potentials by a weighted Lebesgue norm $L^3_{-5/3}$. In
particular, we enhance the weight exponent from $-5$, which was established by
Desvillettes, to $-5/3$. Moreover, we prove that the weighted Lebesgue norm
$L^3_{-5/3}$ is optimal for both exponents.
