No ... i presume you still know which email you used when you purchased LW. Simply use that one & go to the LW support site & downld it again. Your personal downld page is a page you should keep bookmarked.

You can copy your complete incomplete folder over to the new computer. Make sure you include all files & especially the download.dat & bak files which are within the incomplete folder. But replace the new incomplete folder after installation & whilst LW is CLOSED! (Grandpa beat me this time! lol )