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Athena swears the Stygian oath. According to Hesiod, every god who swears falsely by the river Styx is punished by being plunged into some kind of coma. He or she will be lying flat, without voice and without breathing, for one year. Then he/she will be cut off from the assemblies of the gods, forbidden to take part in their counsels and festivals for nine years (and this is described as an even harsher trial than the coma) before he/she is accepted again.

There is a scene in the Iliad where Hera swears the Stygian oath and lies. She assures Zeus that she has nothing to do with Poseidon helping the Greeks against Zeus's command (but in fact she made it possible by seducing Zeus and divert his attention from the battle field. It was her plan all along). It is a bit unclear whether Zeus knows that she's lying or not, but she's never punished for it. He just says something like: "If this is true, then go and do what I tell you".
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