Question

Who was the son of greek god Zeus ?

Answer 1

Zeus had numerous sons in Greek mythology like NUMEROUS . He really g0t around . Greek gods based on the myths had typical powers/abilities of having/granting immortality , super strength , eternal youth , gift of granting boons/curses , shape shifting , mind control , flight etc. Since you specified how they were worshipped I will list only his fully divine sons a :
1. Ares - his son with Hera , his sister-queen . His most prominent lover was Aphrodite , the goddess of love , beauty and sexuality . He was the god of war specifically representing the dark , untamed and violent aspects of war .
2. Heracles - who became a god upon his death . He was the greatest of the Greek heroes seen as the epitome of manliness and virility by the Greeks . He was married upon entering Olympus to Hebe , the goddess of youth , the cup bearer of the gods , who was the daughter of Zeus and Hera .
3. Apollo - his son with his Titaness lover , Leto . Twin brother of Artemis , the virgin goddess of the hunt . He was the god of prophecy , light , music , poetry etc.
4. Perseus - his son with the mortal Argoan princess , Danae . Made into an immortal constellation upon his death . Saved Andromeda , the princess of Ethiopia , from being devoured by a sea monster , whom he married afterwards . Slew the only mortal Gorgon , Medusa .
5. Dionysus - son with the demigoddess Theban princess , Semele . the Greek god of wine and revelry . Husband to Ariadne , the princess of Crete , who helped Theseus escape the monster , the Minotaur and who was the daughter of Minos , a son of Zeus and king of Crete and his queen , the sea goddess , Pasiphae .

6. Hermes - son with the Pleiade , Maia , the daughter of Atlas , the Titan condemned to hold forever , the heavens , on his shoulders . He was the messenger of the gods and was responsible for guiding the newly deceased’s souls to the Underworld . He was also the god of thieves , travelers and messengers .

7. Pollux - son with Leda , the queen of Sparta . Half-brother of Clytemnestra , the princess of Sparta and later Queen of Argos as the wife of King Agamemnon , Helen of Troy , the later queen of Sparta by becoming the wife of King Menelaus , Agamemnon’s brother and Pollux , his mortal twin . Later turned along with his brother into the constellation , Gemini .

Answer 2

Heracles is the son of greek god Zeus