Exact values of the function Γ⁎(k)

The function Γ⁎(k) is defined to be the smallest number s such that a diagonal form of degree k in s variables with (rational) integer coefficients is guaranteed to have a nontrivial zero in the p-adic fields Qp for all primes p. In this article, we find the exact value of Γ⁎(k) for k=14, 20, 24, 26, 27, 29, and 31, and obtain partial information about Γ⁎(32). After the results of this article, the exact value of Γ⁎(k) is known for all k⩽31.