The thing about having them is that if you look in your Todo list (left shoulder button) it will list the deathrow prisoners by name with a percentage next to it. That number is their clemency rating which shows the progress of their appeals to keep from being executed. When that number hits 90% or higher they are approved by the state for execution. At that point it is up to you to schedule their execution and to carry it out. If the number isn't changing at all then you need to make sure there is a Parole room nearby and schedule Deathrow Parole hearings in the program window.
Now an execution can be carried out before that number hits 90% but doing so can land you in seriously hot water as someone you execute too early might be later found to be innocent or there might be circumstances that commute their sentence and make them max security prisoners instead. So check to see where they stand. Might be time to give "Ol' Sparky" a warmup. 
