This paper gives an abstract sufficient condition on Riesz basis with parentheses property for the generators of C-0-groups in Hilbert spaces whose eigenvalues are comprised of some finite unification of separable sets after taking the algebraic multiplicities into account. The condition is then applied to the closed-loop system of a serially connected string system under joint damping feedbacks to show that there is a family of generalized eigenfunctions that form a Riesz basis with parentheses in the state space. The spectrum-determined growth condition is concluded as a consequence.