The gross theory of nuclear β-decay developed by Yamada, Takahashi and Koyama for allowed transitions is extended to include the first forbidden transitions. The effect of the giant-resonance is taken into consideration by using sum rules. The β-strength function of both allowed and first forbidden transitions and the β-decay half-lives are calculated. It is shown that the theory can explain the gross properties of the first forbidden transitions as well as the allowed ones. In particular, the theoretical β-strength functions of neutron-deficient heavy nuclei are compatible with experimental data recently obtained by Hansen et al. It is also shown that the first forbidden transition often predominate over the allowed transition in heavy nuclei.