An identity concerning the Riemann-zeta function

For a certain function $J(s)$ we prove that the identity $$\frac{\zeta(2s)}{\zeta(s)}-\left(s-\frac{1}{2}\right)J(s)=\frac{\zeta(2s+1)}{\zeta(s+1/2)},$$ holds in the half-plane Re$(s)>1/2$ and both sides of the equality are analytic in this half-plane.