No inscription or chronicle records when Hammurabi was born. What survives fixes only his reign, conventionally 1792 to 1750 BCE under the Middle Chronology, and everything about his life before the throne is unrecorded.
What would resolve this A dated Old Babylonian document naming Hammurabi's age or birth before his accession, distinct from the reign dates fixed by his own year names and later king lists.
OpenOld Babylonian history, Mesopotamian chronologyJeremy Black and Anthony Green, Gods, Demons and Symbols of Ancient Mesopotamia: An Illustrated Dictionary