If we have an closed operator $L$ which is selfadjoint, densely defined and strictly positive in the Hilbert space $X$, then how to define $L^{s}$ for any real $s$. I just know its $L^{s}$ for $s \in \mathbb{N}$.
I dont know about any reference from where i can study these. Please help me
You may have a look e.g. at Thm 3.1 (p 488) in S. Lang, real and functional analysis. But the spectral theorem may be found in many books (I think that wiki is not very good on that, though).
Given an unbounded selfadjoint operator $A$ and a real valued Borel measurable function $f$ on ${\Bbb R}$ (in your case that would just be $t\mapsto t^s$ for $t\geq 0$ there is a unique self-adjoint operator $f(A)$ such that the domain consists of those $v\in H$ for which $f\in L^2(\mu_v)$ (where $\mu_v$ is the spectral measure associated with $v$ and $A$, in your case the support is on ${\Bbb R}_+$).
The action on such $v$ in the domain verifies e.g.: $$ \langle f(A)v,v\rangle = \int f d\mu_v .$$