The serial number on your Omega watch will usually be found either laser etched in tiny digits on the back or for older watches it may be on the inside of the case. It will be a number (no letters) between 5 and 9 digits long. The number containing dots, spaces and possibly letters is not the serial number, that is the model or case reference number which can be decoded using the Omega watch model decoder.
You can also look up the date from the Omega movement/caliber number that is etched directly on the movement itself using the Omega movement database.
When combined with the information from the other tools you can get help to validate an Omegas authenticity or to check if you have a fake Omega.This tool can date vintage Omegas from 1895 to 2008.