Eigenvectors of half-derivative operator

The eigenvectors of the derivative operator $$D$$ are $$\exp(kx)$$, as talked about in this post and computed with Wolfram Alpha here.

The eigenvectors of $$D^2 = D \circ D$$ with eigenvalue -1 are linear multiples of sine and cosine (with the argument in radians), computed with Wolfram Alpha here.

What are the eigenvectors of the half-derivative operator $$D^{(1/2)}$$ and is to possible to even phrase the question as a differential equation?

To address a potentially relevant concern raised in the comments of this question, I don't know exactly which definition of fractional derivative I want, I'm hoping that the collection of eigenvectors of the operator itself is a property where all or most reasonable definitions of the fractional derivative agree.