Hello, It will be my pleasure to assist you with your computer today. I understand how this could be frustrating to not have your serial number for Adobe Lightroom 3. However, if you purchased Lightroom, then there is definitely a corresponding serial number somewhere.If you purchased it as a download on Adobe.com, go to Adobe.com and click on in the Sign In link in the upper-right area. Sign in with your Adobe account and then click on the Welcome menu and select My products and services from the drop-down. Click on the triangle next to each program to see the serial number.If you purchased Lightroom 3 on CD, which is more likely the case given the age of the program, then the Serial Number came printed on the CD Sleeve.
If you're unable to find the original CD sleeve, you can purchase a discounted copy of Lightroom 3 from here.