I don't think anything is much of a deterrent, because that overestimates the extent to which criminals are rational agents. But there is a massive difference

I don't think anything is much of a deterrent, because that overestimates the extent to which criminals are rational agents. But there is a massive difference between life imprisonment and the death penalty – with life imprisonment, there is always the possibility of some day getting out due to a commutation, a change in the law, etc. Whereas, once someone is executed, they are dead.