Can I legally shoot a dog that is in my yard growling at my children and killing my livestock?

This is not my dog and we have asked the owners to please keep it on their property. They will not contain the dog in any way, They have brothers who work at the local police department so they will not even come out and talk to us. We have called animal control and the pound and they will not help as well. I'm out of options. This dog has cornered me and my children in our car jumping on the doors and barking/growling. More times than I can count it has made us run from our fenced in yard to the house before he could bite us. He has killed nearly 20 of our animals and run our cows through the fence. Yesterday was the last straw it grabbed one of our turkeys and tore it to shreds before I could hit him enough times with a shovel to make him stop. I ran inside with the turkey and by the time I came back outside the dog was back inside its house. I'm sure the neighbors heard me screaming or maybe even saw what happen. I'm sick and tired of being threatened and finding the leftover pieces of my animals on my lawn every few days.Just to answer the questions before you ask... Yes my property is fenced 100% and he digs under the fence at night or jumps it if we're outside. He is a mixed breed around 120 pounds and we live in Utah.