If you took the time to register your original installation with Adobe (which most people don't do) you could log into your Adobe account and obtain the serial number. If you are using a Windows computer, and still have access to the old computer where Lightroom is installed you can do the following:
1. In the Windows folder options ensure that you are able to see hidden folders.
2. Go to the folder C:\ProgramData\Adobe\Lightroom.
3. Use Notepad to open the file, Lightroom 5.0 Registration.lrreg
You will be able to retrieve your serial number from that file.
If you took the time to register your original installation with Adobe (which most people don't do)
Yes, this is sad but true, and most people don't take the time to make a note on paper or digitally of their serial numbers either, and everyone would be better off if people did this without fail.
I make a text document with all the serial numbers of software that I have purchased, and that document is backed up locally and in the cloud. (Hint: for anyone reading this, I recommend you do this too)
Thanks that was easy.
While i registered all my cs2 thru 6 versions and my lightroom 2 thru 4 versions and recorded the serials and saved the adobe registration emails. Since this came as a freebee with the camera i neglected this vital step. I have the instslled version on my old laptop but want to install it on my new one. Thanks to Jim Hess I got the serial so all is well in my world and I will register it with ADOBE