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Was the F-104G the first production fighter with inertial navigation?
Short answer
With caveats: Litton's LN3-2A was the first inertial navigation system small, light, and accurate enough to fit in a high-performance fighter. It was the F-104G's development, around 1959, that brought it to the aircraft. Boeing's own history credits the REINS as the first INS on a production aircraft.

Litton's LN-3
The LN-3 was an inertial navigation system developed in the 1960s by Litton Industries. Its LN3-2A variant was the first inertial navigation system small, light, and accurate enough to fit into a high-performance fighter. The unit ended up installed on the versions of the Lockheed F-104 Starfighter used as attack aircraft by European air forces.
From contract to 2,000 units
Around mid-1956, Wright Air Force Base awarded Litton Industries a contract worth roughly $300,000. The system was finished and ready for flight testing by late 1958. Based on those results, Litton Industries received a contract to supply 2,000 systems for NATO's F-104.
What exactly was the first
It was the development of the F-104G, around 1959, intended for European air forces with tactical bomber and strike capability, that introduced the LN-3 into the aircraft. The LN3-2A's pioneering status refers to the size, weight, and accuracy required for a high-performance fighter. Boeing's own history, by contrast, holds that the REINS was the first inertial navigation system on a production aircraft. The LN-3 thus remained tied to the attack variants of the European Starfighter.
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