You don't mention which product you registered..if it's older..the serial number is probably for Live Lite 9.
You have to download Live Lite 9 through the Ableton archives.
I just checked/downloaded and ALL versions of Live Lite are still available in the Archived downloads.Last version Live 9 Lite Version 9.7.7.
Here is the link to the Archives , if you already have an account with Ableton, just login and download..The drop down menu has all versions 64 bit, 32 bit Windows and Mac.Download/install with the serial number you have.
If the serial number you have is for Live Lite 10..it's in there also..