As far as I know, LimeWire does not use a serial number. Perhaps you need to post your question to a forum that deals with your question more specifically. However, the company who made the software should be handling the issuing of serial numbers.

If you have downloaded some software that normally requires you to pay for a serial number, most likely there will also be someone out there with a serial number, probably in the form of a text file. I should not have to tell you though, that this is not likely what the company intended for you to do.