One more remark, I found a note claiming only about 50,000 FL pistols were made during the war, all for the German army. So the high serial number and lack of imperial markings indicate a post WWI Weimar Republic production (they continues making them throughout the 20s, supposedly assembling some from parts until 1936.
Edit. Promptly found another article
stating all that is nonsense, the gun was only built until 1918, but about 10 - 15% of production where civilian and have no imperial markings. Also gives serial number ranges for the different sub-models of the FL.