This controversy has generated debates is philosophical circles as well as in criminology and in international diplomacy. The arguments can be condensed to whether the death penalty is a just punishment, whether it is effective as a preventative, whether it is a murder by the state, whether it violates the Constitution's ban on "cruel and unusual punishment", whether it is painless, and whether it is actually a financial budget decision (because life sentences are expensive for the state). Add to that the irreversability (especially now when DNA evidence is exposing so many wrongful convictions).
You'll find that the main divides lay in some people supporting capital punishment, while others opposing against it.
Those for capital punishment see it as a just means to prevent further crimes, as they claim other 'criminals' will be deterred from committing crimes, if they know the punishment is severe.
Then again, there are those that condemn capital punishment, and regard it an inhumane way to 'punish' and treat people. They also argue that perhaps the verdict was wrong, and so the 'criminal' will be faced with a harsh punishment for a crime they didn't do.