Fractional Laplacian in higher order case

Question

Let $n\geq 2$ and $\sigma \in (0,\frac{n}{2})$. Denote the critical Sobolev exponent $2_{\sigma}^*:=\frac{2n}{n-2\sigma}$, consider Sobolev space $E$ which is the space of real-valued functions $u\in L^{2_{\sigma}^*}(\mathbb{R}^n)$ whose energy associated to $(-\Delta)^{\sigma}$ is finite, that is, \begin{equation}\label{norm1} \|u\|_E^2:=\int_{\mathbb{R}^n} u(-\Delta)^{\sigma} u \,\mathrm{d}x=\int_{\mathbb{R}^n}|x|^{2\sigma}|\widehat{u}|^2 \, \mathrm{d} x<+\infty, \end{equation} where $\widehat{~}$ denotes the Fourier transform. The space $E$ can be also seen as the completion of the space of smooth function with compact support in $\mathbb{R}^n$ under the norm \eqref{norm1}. Note that, if $\sigma=1$, then $(-\Delta)^{\sigma}=-\Delta$ is the classical Laplacian operator, and \eqref{norm1} becomes $$ \|u\|_E^2=\int_{\mathbb{R}^n}|\nabla u|^2\, \mathrm{d} x, $$ and if $\sigma \in(0,1)$, then $$ \|u\|_E^2:=\int_{\mathbb{R}^n\times\mathbb{R}^n} \frac{[u(x)-u(y)]^2}{|x-y|^{n+2 \sigma}}\, \mathbb{d}x \, \mathbb{d} y, $$ My question is when $\sigma>1$, do we have an explicit expression of $$\int_{\mathbb{R}^n} u(-\Delta)^{\sigma} u \,\mathbb{d} x.$$