Provided you registered the software when you purchased it, you don't need the serial, just log in to the account you used to register or purchase the software, and visit this page:http://fxhome.com/account/programs
Download the latest installer, run it, and then when the activation wizard appears, select the "Use my FXhome account details" option, and you'll be good to go.
If you didn't register the software when you bought it, then that was unwise, and you may be out of luck. How long ago was this? Did you purchase EffectsLab Lite (discontinued years ago) or EffectsLab Pro? There doesn't appear to be any software registered to the account you are currently logged into, so make sure you use the correct account when you try to activate.
If you can't remember the account info, submit a support ticket, as only the folks at the office will be able to track that info down for you.