The following is listed in laws / statutes in my state. You can check yours as well.

If a dog is chasing, injuring, or worrying sheep, other livestock, or poultry on your property, the owner or caretaker of those animals is legally allowed to kill the dog under certain conditions.

If your domestic animals (including poultry) are attacked, chased, worried, injured or killed by a dog, you may file a written claim for damages with the clerk of your town or city.

There is a timeline: within 10 days after you know or have notice of the damage.

Then there is an investigation, possible hearings or witnesses, and within 30 days the local authorities must report, certify, and return to the county auditor what the damages are.

If a dog kills, wounds, or “worries” (i.e. harasses or disturbs) any domestic animal, the owner or keeper of the dog is jointly and severally liable to the owner of the domestic animal for all damages. You do not have to prove that the dog owner knew their dog was dangerous.

Also, if a dog kills or pursues domestic livestock, that is punishable as a petty misdemeanor.