This website is a good resource for the animal laws of various states. I found this, specific to Alabama, there:
(Acts 1915, No. 185, p. 259; Code 1923, § 3221; Code 1940, T. 3, § 5.)
§ 3-1-6. Liability of owner, etc., for injuries to livestock, etc., caused by dog while off premises of owner, etc.
If any dog, not being at the time on the premises of the owner or person having charge thereof, shall kill or injure any livestock, the owner or person having such dog in charge shall be liable for damages sustained by the killing or maiming of any livestock and for the full costs of the action.
§ 3-1-6. Liability of owner, etc., for injuries to livestock, etc., caused by dog while off premises of owner, etc.
If any dog, not being at the time on the premises of the owner or person having charge thereof, shall kill or injure any livestock, the owner or person having such dog in charge shall be liable for damages sustained by the killing or maiming of any livestock and for the full costs of the action.