This is a user to user forum. We are all users like yourself and have no connection with Adobe. Unless you are using an old out of date version of Lightroom, there is no serial number. The current versions of Lightroom are installed/activated using your Adobe ID and password. What version number of Lightroom are you referring to?
If you are in fact using an old out of date standalone version of Lightroom, and need your serial number, it will be necessary for you to contact Adobe. Here is a link where you might be able to obtain such assistance: