The full (and official) Luger serial number appears on the front of the grip frame under the barrel. The serial number on other parts may be incomplete.
Military Luger serial numbers ran 1-9999, then 1a-9999a, 1b-9999b, and so on. Numbering began with 1 at the start of each year. So a complete and unique Luger description must include the manufacturer, the year, and the full serial number including the suffix letter if any.