You don't need a new serial number, you need your previous serial number to use, again. You can only use the redemption code, once, and need to have recorded your serial number, somewhere.
The usual place to get your old serial number back is by logging into www.adobe.com and clicking Manage account then a few more clicks into your product lists and you'll see all the serial number for your registered products.
If you never registered your LR copy with Adobe then they won't have your serial number. I don't know if there's a way to get it back or not. You can try doing the help and support option on adobe.com and get to where you Still Need Help and do the Chat option and see if they have your old serial number stored somewhere even if you never registered it.