The Iliad provides two moments where some type of institutional (non-violent) conflict resolution is imagined. Note, though murder is mentioned, capital punishment is not. And Homer’s world is certainly a savage one…
At the end of the embassy in book 9, Ajax voices his frustration with Achilles (9.632-638):
“You are relentless: someone might even accept payment
for the murder of a brother or the death of his own child.
and after making great restitution, the killer remains in his country,
and though bereft, the other restrains his heart and mighty anger
once he has accepted the price. But the gods put an untouchable
and wicked rage in your heart over only a girl…”
νηλής• καὶ μέν τίς τε κασιγνήτοιο φονῆος
ποινὴν ἢ οὗ παιδὸς ἐδέξατο τεθνηῶτος•
καί ῥ’ ὃ μὲν ἐν δήμῳ μένει αὐτοῦ πόλλ’ ἀποτίσας,
τοῦ δέ τ’ ἐρητύεται κραδίη καὶ θυμὸς ἀγήνωρ
ποινὴν δεξαμένῳ• σοὶ δ’ ἄληκτόν τε κακόν τε
θυμὸν ἐνὶ στήθεσσι θεοὶ θέσαν εἵνεκα κούρης
On Achilles’ shield, Hephaestus creates two cities, one at war and one at peace. He also creates an assembly where a killing is being judged Il. 18.496-508
“The people where gathered, crowded, in the assembly where a conflict (neîkos)
had arisen: two men were striving over the penalty for
a man who had been killed; the first one was promising to give everything
as he was testifying to the people; but the other was refusing to take anything;
and both men longed for a judge to make a decision.
The people, partisans on either side, applauded.
Then the heralds brought the host together; the elders
sat on smooth stones in a sacred circle
as they held in their hands the scepters of clear-voiced heralds;
each one was leaping to his feet and they pronounced judgments in turn.
In the middle there were two talents of gold to give
to whoever among them uttered the straightest judgment.”
λαοὶ δ’ εἰν ἀγορῇ ἔσαν ἀθρόοι• ἔνθα δὲ νεῖκος
ὠρώρει, δύο δ’ ἄνδρες ἐνείκεον εἵνεκα ποινῆς
ἀνδρὸς ἀποφθιμένου• ὃ μὲν εὔχετο πάντ’ ἀποδοῦναι
δήμῳ πιφαύσκων, ὃ δ’ ἀναίνετο μηδὲν ἑλέσθαι•
ἄμφω δ’ ἱέσθην ἐπὶ ἴστορι πεῖραρ ἑλέσθαι.
λαοὶ δ’ ἀμφοτέροισιν ἐπήπυον ἀμφὶς ἀρωγοί•
κήρυκες δ’ ἄρα λαὸν ἐρήτυον• οἳ δὲ γέροντες
εἵατ’ ἐπὶ ξεστοῖσι λίθοις ἱερῷ ἐνὶ κύκλῳ,
σκῆπτρα δὲ κηρύκων ἐν χέρσ’ ἔχον ἠεροφώνων•
τοῖσιν ἔπειτ’ ἤϊσσον, ἀμοιβηδὶς δὲ δίκαζον.
κεῖτο δ’ ἄρ’ ἐν μέσσοισι δύω χρυσοῖο τάλαντα,
τῷ δόμεν ὃς μετὰ τοῖσι δίκην ἰθύντατα εἴποι.