How do I solve for the order of integration in the Fourier transform-based Differintegral?

Question

In a comment response to this post someone brought up the Fourier transform-based Differintegral, which is defined as follows: $$f^{(s)}(x)=\frac{1}{2\pi}\int_{-\infty}^{+\infty} e^{- i \omega x}(-i \omega)^s \int_{-\infty}^{+\infty}f(t)e^{i\omega t}dt \, d\omega$$ How can I solve for $s$ (which is the order of differentegration) based on this definition?