I believe that the judge told Zak that he couln't appeal his guilty plea.
Maybe he can appeal the sentence and end up with death next time.
Sadly, there MAY be sufficient grounds for an appeal to succeed thanks to the record of proceedings indicating the loon has, arguably at least, a mental illness. In truth, he is clearly an evil, heartless, murdering coward. And remember the tactical errors of the prosecution team which almost blew out the case before the jury reached its decision.
Unfortunately, the death sentence option is out the window. Subplot pilot could get no more than what he already received, life in prison without the possibility of parole.
Dang it.