I knew you'd bring that up - three things.
1# It's virtually impossible for it to be a medieval fake, unless it was created from the original by an artist with unheard of skill. In particular - the shroud depicts a man crucified in the manner appropriate to the 1st century AD, not in the manner the medieval Christians believed Christ was crucified. Also, there's the problem of the blood-spatter matching exactly to the face cloth which is known to be older.
2# The Shroud is reported in Constantinople at least a century before 1350.
3# It's now believed the carbon-dated strands were erroneously taken from a medieval repair to the cloth, contaminating the results.
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