For a prime p with p≡3 (mod 4) and an odd number m, the Bentness of the p-ary binomial function fa,b(x)=Tr1n(axpm-1)+Tr12 is characterized, where n=2m, a ∈ F*pn, and b ∈ F*p2. The necessary and sufficient conditions of fa,b(x) being Bent are established respectively by an exponential sum and two sequences related to a and b. For the special case of p=3, we further characterize the Bentness of the ternary function fa,b(x) by the Hamming weight of a sequence.