A corporate jet Captain reported losing FMS shortly after takeoff resulting in distraction and a near stall above FL400.

Narrative

Three minutes into flight; there was a 'L6 Key Stuck' message on CDU-1. That was preventing CDU-1 from further use. Selected CDU-2 reversion switch and realized that CDU-2 could not access FMS functions (single FMS aircraft). Notified ATC that was unable waypoint/RNAV navigation until further notice and requested vectors while I determined course of action. Was cleared to and continued climb to FL230 where we briefly leveled. [I] was then cleared to FL410. Climb initiated 1500FPM. [I] began bringing out commercial charts to request full airway clearance (VOR routing) from ATC as I would no longer be able to use FMS-1. Developed full route request and relayed same to ATC as we were climbing through FL360. As I was folding charts back up; what initially felt like an abrupt auto-pilot disconnect and pitch down occurred (145KIAS near FL405) certainly due to stick pusher. That was followed within five seconds by pitch up to prior climb attitude. This cycle repeated once more. I disconnected auto-pilot via yoke disconnect after first cycle; but pitch trim was still in nose up condition. It required about five seconds for me to correct trim to establish level flight at FL397. Requested block FL380-FL410 from ATC and was granted. After six or seven minutes; notified ATC that diversion at FL280 was desired. In short; the entire event was a function of my leaving the VS mode of the FGC at +1500 while I was researching airway routing and distracted. This aircraft type doesn't climb at +1500 well above FL340 and a severe airspeed loss was the result. All three airspeed indicators/altimeters were in full agreement at all times.