You can locate the 17-digit Serial number at these locations.
Back by the ports on a small sticker with a barcode (green), separate from the main sticker (red).
It may be underneath the unit by the feet, depending on the model.
If the projector is still running you can find the serial number on the info/about tab
If you are looking to find the Model of your unit you can use this: Optoma Model # Locator.