Find your Serial Number of a Mac.

First off, do not post any serial number on Apple Support Communities. The Serial Number is your personal key to your warranty on your computer. The only thing that will be needed is the model name, year, and time of year of Mac for troubleshooting most issues that are resolved on this board.

You want to know your serial number? Apple has this article below:

Find the serial number of your Apple product - Apple Support

Knowing your serial number, the age, and model name of the machine is usually able to be revealed on https://checkcoverage.apple.com/

When it won’t boot the operating system the entire way, a magnifying glass

is sufficient looking on the underside of the machine. The zoom feature

of cell phone cameras or regular point and shoot cameras with good light can also reveal it.

If your Mac will only boot into Recovery, open the Utilities menu -> Terminal and type:

system_profiler SPHardwareDataType

followed by the enter key. Macs that shipped with an installer CD/DVD for Mac OS X, can also boot with the Option key on the installer, and have a Utilities menu -> Terminal where that command can be run.

Short of these solutions, it may be necessary to bring to an authorized service provider. Sometimes Macs after a logicboard repair get a new serial number, which should be flashed onto the board, and updated through Apple's databases by an authorized service provider. If it isn't, that's where the past repair location will need to use their diagnostic tools to figure out the serial number. You may find the Profiler has an empty serial number listed, in which case, it may be one of these forgotten flash solutions.