An identity concerning the Riemann-zeta function

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