There's been masses of research in this. Death penalty isn't a deterrent for anything. The stiffness of the penalty is never a deterrent for any crime. Nobody commits a crime if they think they might get caught. It's pretty basic. The only real deterrent is the perceived chance of getting caught.
Having the "death penalty as deterrent" in a political program has always been a rhetoric/propaganda tool to win over stupid voters. Don't be one of them. If you fall for it you are encouraging politicians to treat us even more like idiots. Not that I can imagine how it could get any worse, but you never know.
It's also arguable that serial killers, like Bundy have a brain damage. As the theory goes, they all have the same kind of frontal lobe damage. Isn't it a bit like sentencing someone to death because they have a deadly disease?
I don't really have an opinion about the death penalty, or any punishments. Science shows that all criminal punishments are a complete waste of time. The more we punish and the harder we punish, the more criminals we get, (unless they're executed he he). It's a lose-lose. I off-course realize that it's good having a criminal system and putting them away. For revenge if nothing else. Always nice to see somebody who've hurt you suffer. But I do also realize that it is pointless. All efforts must be made in preventing the crime being committed in the first place. Once it's been carried out it's too late. And the longer we put the criminal away, the larger the chance he/she will commit it again.