What is the eigenvector of the functional square root of the differentation operator?

Question

The eigenvector of the differentation operator is the exponential function.

But I wonder, what are the eigenvectors of operators in fractional calculus?

For example, the "functional square root" of the differentation operator, D: https://wikimedia.org/api/rest_v1/media/math/render/svg/a09bb2792735635495d9638374cea75b81f37fb1