You can open System Profiler and see if it contains your serial number.
Or have the computer near a previous connected Wifi or on a Ethernet cable to your router.
Reboot holding the D key down for like what would appear to be 10 minutes until Hardware Test appears on the screen.
You can run a test if you wish.
There is a pane at the top if selected will give you your new serial number.
You also can use Terminal
You can quickly retrieve any Macs serial number via the command line by using the following command string:
system_profiler |grep "r (system)"
The command results will look something like this:
$ system_profiler |grep "r (system)"
Serial Number (system): C24E1322XXXX
If that fails you can give them your MAC address which is unique to each machine and there are records at Apple with it.
Open Utilities > Terminal. Copy, paste and press enter:
For your computers Ethernet (wired) MAC address: ifconfig en0 | grep ether
For your computers Airport (wireless) MAC address: ifconfig en1 | grep ether