Where did you download it from?
- If you bought it at the App Store, there is no serial number. If Aperture is asking for a serial number, someting is wrong.
- If you bought a serial number from the Apple Store (before the App Store was introduced) to unlock the Trial Version, you should have received an email with the serial number. Search your mail box.
If you cannot find a serial number, contact Apple for a replacement serial number.
See this link: Pro Application Replacement Serial Numbers
Do you have a second mac, where Aperture is still running? You can see the serial number in the "About" panel: In the Aperture menu in the main menu bar: Aperture > About Aperture.