This question is being discussed all over the world with different conclusions. As it has been already said:
1. death penalty is an absolute penalty which cannot be remedied - in case when person who was sentenced will later be found innocent (so called "lapse of justice"). These things happen sometimes intentionally (when the government wants someone to disappear) or accidentally (when judge makes a mistake or doesn't have all necessary information)
2. death penalty prevents pathologic criminals from commiting another crime - this can be achieved as well by a life imprisonment in isolation
3. death penalty is cheaper for the state than life imprisonment - there were calculations which neglected this - using an electric chair is not so cheap
4. right to live is a fundamental right of every human being - you don't have right to kill anybody. Doing this as a punishment for murders is like becoming the same murder as the sentenced person
5. there were situations when the person sentenced to death penalty changed his personality and was behaving as a "good prisoner". I don't say that he should be set free in this situation but he can do lot of good things in his future life (despite being in prison). I've seen beautiful paintings and sculptures made by prisoners. By killing this person, you neglect them an opportunity to change themselves and become self-possesed
I've talked about this problem with a professor of penology (science about execution of punishments, related to criminal law) who is a former prosecutor. He was present at several executions of death penalties and said that it were the worst moments in his life. When observing this, he realized that there is nothing useful about this punishment. He told me one more thing: "As a civilization of humanity, we must believe that every person is in his deep essence good and there is always a chance that the criminal will find his inner peace one day. Until this day we have to keep him out of society, talk to him and teach him how to be good. If nevertheless all this fails, we still do not have the right to kill him, because we will become the same villains as we consider him, and what is more, we would resign from the very basic idea of humanity which forms our society." I believe his words are sincere and wise. I had a chance to visit several prisoners and this changed my attitude to punishments in general.
You will never understand what does it mean to be sentenced for a year imprisonment (which appears to be really minor punishment) until you spend an hour or a day in prison