I think you need to know what “deter” means. Deterring someone who has already committed the crime and is physically prevented from repeating the crime is kind of pointless, don’t you think?
The point of using capital punishment as a deterrent is to serve as a warning to others who might plan to commit similar crimes, in an effort to discourage that. However, it doesn’t work. If it did, Texas and Florida would have the lowest murder rates in the country, as they execute the most people, wouldn’t they? And they don’t. Not even close.
Also worth mentioning — the five states with the lowest murder rates (in 2014, the most recent year with complete data I found) are New Hampshire, Maine, Minnesota, Vermont, and Hawaii. Maine, Minnesota, and Hawaii have abolished capital punishment completely. Vermont has abolished it for state crimes (as murder usually is), but Vermont residents convicted of certain federal crimes in a federal court (such as murder or treason) can be subject to the death penalty. The only state of the five with capital punishment still on the books is New Hampshire. New Hampshire has carried out only 24 executions since 1734, the last one being in 1939. They don’t even currently have an execution facility, so they appear to be, for all intents and purposes, a non-capital punishment state.
So if you have some kind of evidence showing that capital punishment is an effective deterrent to crime, I’d like to hear it. Only don’t insult my intelligence with some variant of “an executed killer won’t kill again”. Putting someone in a maximum security prison for the rest of their life prevents them from murdering, too.