Following a review of tracking data, and consultation with experts, it is believed Hamlet was shot during the first week of January.
It is thought he was injured in an area between Stanhope and Glenbreck off the A701 Broughton to Moffat road.
Det Sgt David Lynn said: “It is extremely disappointing that we are investigating another golden eagle persecution crime.
“Very few people would have the means, opportunity and motive to commit this crime and I would urge anyone with any information to contact us, external so we can identify whoever is responsible for shooting Hamlet.”
Police are working with the UK National Wildlife Crime Unit and anyone with information has been asked to come forward.
Hamlet was relocated to the area as part of the South of Scotland Golden Eagle Project (SSGEP) which has boosted bird numbers in the area.