The deterrent argument never made sense to me. The fact that someone is contemplating a heinous murder, and they would say to themselves "You know what? I'm willing to risk life in prison, but since I might be put to death, I guess I'll go home and cool off." I could, however, believe that it helps prosecutors get plea deals.
In my personal experience, a vast number of people want revenge killing. Talk to anybody about a child murderer, and they will usually express a desire to see them raped or tortured or killed. That isn't really how we should define our formal justice, in my opinion, and in any event I would think living out your life in prison is far more punishment than simple death.
From a practical standpoint, execution wastes tax dollars. New York state projected that the death penalty costs $1.8 million dollars just through trial and initial appeal. Housing on death row costs twice as much as gen pop prisoners.
So even if we had a perfect process for identifying transgressors, I don't see the point. I also dislike the international club we belong to in this area. India and Japan are probably the most savory on a list that includes North Korea, Iran, Saudi Arabia, and Nigeria. India has practically abolished it, only using it five times in 20 years. The current rankings in executions:
China, Iran, Pakistan, Saudi Arabia, United States. Wow that's a fun group to be a part of. Gonna be hard to move up the list unless we streamline our process like China does. Which some death penalty fanboys have suggested - complaining that the appeals process should be shortened to make it less expensive.