You can get an unambiguous answer, but the answer depends on your assumption:
Under the weakest restrictions, there is substantial ambiguity: we cannot rule out the possibility that having a death penalty statute substantially increases or decreases homicide. This ambiguity is reduced when we impose stronger assumptions, but inferences are sensitive to the maintained restrictions. Combining the data with some assumptions implies that the death penalty increases homicide, but other assumptions imply that the death penalty deters it.
From a new and careful paper. Short answer: we have no idea. But there’s a reasonable chance that it raises homicide rates as much as it lowers them.
If you think (as I do) that the burden of proof ought to be on the proponents of more draconian punishments, well, the burden is not borne.
Of course, like most policy, it’s silly to pretend that this is a debate based on evidence and reason, let alone real deterrence (on either side).