It all depends on what version you have. There is the Stand Alone version that has a perpetual license, Never Expires, and the version that comes with the Creative Cloud subscription, Photoshop + Lightroom. The last one is based on a yearly subscription. If you don't keep up on the payments it expires.
The Stand Alone version can be bought in a box or as a download. If you bought it in a box the license key, Serial Number, is on the packaging, box, someplace. If you bought it and then Downloaded it whoever you bought it from would of sent you an Email with the Serial number
It will only show up under your products if you took the time to register Lightroom. Otherwise, Adobe will have no record of your serial number. What operating system are you using?