We present third-order many-body perturbation theory calculations of the Lennard-Jones C3 coefficient for the alkali-metal atoms lithium, sodium, potassium, rubidium, cesium, and francium. All-order singles-doubles calculations of C3 are also presented for lithium, sodium, and potassium. For lithium and sodium the present values of C3 are compared with other theoretical and semiempirical values...