Download the trial version, install it, use the serial number that you already have. There is nothing for you to "redeem".
You can install LR on up to two computers so you’re not moving LR to the other computer, you’re installing it again on another computer.
For LR to function in non-trial mode on the first computer you would have had to use a serial number to license it.
Are you asking how to find the serial number, because you’ve lost track of what it is, or what do you mean?
Are you going to abandon using LR on the first computer and only use it on the second so you’ll need to move your LR catalog and photos over or is the copy of LR on the second computer going to working with a new catalog and new photos?