Bernstein's constant doesn't strictly fit the parameters of the question, but it's notable as a more recent "Legendre-type" example. Bernstein conjectured that his constant was exactly $\frac{1}{2\sqrt{\pi}}$ in 1914; it wasn't until the '80s that it became possible to compute enough digits to refu(dia)te the conjecture.
Bernstein's constant doesn't strictly fit the parameters of the question, but it's notable as a more recent "Legendre-type" example. Bernstein conjectured that his constant was exactly $\frac{1}{2\sqrt{\pi}}$ in 1914; it wasn't until the '80s that it became possible to compute enough digits to refu(dia)te the conjecture.