On the classical Turán problem for 3-graphs

At the moment, no counterexample is known to Turán's conjecture T(n,5,3) = 2 \binom{n/2}{3} for even n.
Under the assumption that this conjecture is true, we find the exact values of T(n,5,3) for odd n.
With the exception of n=5,7,13 and possibly n=27, the answer is equal to the minimum number of monochromatic triangles in a 2-edge-colored K_n.