If you took the time to register your initial purchase of Lightroom with Adobe then you can retrieve the serial number from your account.
I did register….. but the Email address is now defunct….Any other smart ideas ?
You can try starting a live chat with someone from Adobe and see if they can help you. But I question whether you'll get very far because they don't support old versions. Just keep following the options on this link, and you'll be able to start a chat session.