Sanhedrin 47B

הרוגי מלכות להרוגי בית דין? הרוגי מלכות, כיון דשלא בדין קא מיקטלי, הויא להו כפרה. הרוגי בית דין, כיון דבדין קא מיקטלי, לא הוי להו כפרה.

those referred to in the verse in Psalms who were killed by a heathen government to those inhabitants of an idolatrous city who were killed by a Jewish court? Those who were killed by a heathen government, since they were killed unjustly, achieve atonement through their deaths, even without repentance. But those who were killed by a Jewish court, since they were killed justly for their sins, do not achieve atonement unless they repent.

תדע, דתנן: לא היו קוברין אותו בקברות אבותיו. ואי סלקא דעתך, כיון דאיקטול הויא להו כפרה, ליקברו! מיתה וקבורה בעינן.

The Talmud adds: Know that this is true, as we have learned in the Mishnah: They would not bury the executed transgressor in his ancestral burial plot. And if it enters your mind to say that once the transgressors are executed by the court, they achieve atonement, let them be buried in their ancestral burial plots. The Talmud rejects this argument: An executed transgressor cannot be buried in his ancestral burial plot, because in order to achieve atonement, both death and burial are needed. But once he is buried in the graveyard set aside for the wicked, he achieves full atonement.

מותיב רב אדא בר אהבה: לא היו מתאבלין, אלא אוננין, שאין אנינות אלא בלב. ואי סלקא דעתך, כיון דאיקבור הויא להו כפרה ⁠– ליאבלו!

Rav Adda bar Ahava raised an objection to this from another ruling in the Mishnah: The relatives of the executed transgressor would not mourn him with the observance of the usual mourning rites, but they would grieve over his passing, since grief is felt only in the heart. And if it enters your mind to say that once the executed transgressors are buried they achieve atonement, let the relatives of the executed transgressor mourn him as well, as he has already achieved atonement.