Not possible to locate online using the serial number.
What happens when you try to use Find My to locate the phone? (We assume that Find My was enabled on the phone.....before.....it was lost.)
Find My is not going to work if the battery has run down or the phone does not have an Internet connection where it might be currently located. All that you can do is keep trying to see if the status of the phone has changed.
Unfortunately, thieves are very fond of putting stolen phones in a special kind of bag that will not allow the phone to send or receive signals.
If your iPhone, iPad, or iPod touch is lost or stolen - Apple Support