Did you register your copy of Lr with Adobe; this is the easiest way of recovering your serial number?
Did your copy of LR come in a box and one a DVD disc or did you download it?
If it came in a box and on a DVD then the serial number is on the packaging.
If you downloaded it then you should of received an Email with the serial number and you should of saved that somewhere. Made a hard copy of the number.
If it was bought directly from Adobe as a download then it should be listed in your Adobe ID.