The paper presents asymptotic analysis of an eigenvalue problem for the Helmholtz operator in a periodic structure involving split-ring resonators [originally proposed in J. B. Pendry et al., IEEE Trans. Microwave Theory Tech. 47, 2075 (1999)]. The eigensolutions are sought in the form of Bloch waves. The main emphasis is given to the study of localized modes within such a structure and to the control of low-frequency bandgaps on the corresponding dispersion diagram. Asymptotic results are explicit and are in good agreement with numerical simulations.