Please click on the below link and follow the steps to find the serial number.
> sign in with your email address (Adobe ID) and password.
> Choose Plans & Products > View all > View your products.
Hope this helps.
If using LR 5 or earlier do a Search your hard drive for a file with extension .lrreg. When found Open the file using Windows Notepad or Mac TextEdit.
You might also want to, in the future, start storing your serial numbers for all products, not just Lightroom, in a place where you can find them again in the future. For example, a piece of paper, or better yet a text file named something like serial_numbers.txt, which you can store on your computer and backup into the cloud (for example, DropBox or a similar service).