Fermi-Dirac half-order integral at zero argument

Expanding the Fermi denominator as an alternating exponential series gives $F_{1/2}(0)=(1-2^{-1/2})\zeta (3/2)\simeq 0.765147$. Thus a band with chemical potential pinned exactly at its lower edge has $n=F_{1/2}(0)N_c$, rather than the classical result $N_c$.