Using Lightroom via CC doesn't use a serial number. I think you have downloaded a standalone version and have installed it. You need to install Lightroom from the creative cloud application manager.
If you do have the CC subscription for both LR and PS just close LR, Open the Creative Cloud app and it should authenticate your install as valid and then give you access to it. If in the CC app it still lists an Update for LR click on that to install from the CC app and you should be good to go.