jmc

By Vieta's formulas, the sum of the three roots is $52.$ If all three roots are odd primes, then their sum is odd, a contradiction; therefore, $2$ must be one of the roots of the polynomial.

Now we can find $k$ without having to figure out the values of the other two roots, because for $x=2,$ we have $$2^3-52(2^2)+581(2)-k=0.$$Solving for $k,$ we get $k=\boxed{962}.$