A polycrystalline sample of La1.5Sr0.5NiO4+δ was prepared by using a solid-state reaction. X-ray diffraction proved the sample to be a single phase with a tetragonal structure (space group: I4/mmm). By using an iodometric titration method to determine the non-stoichiometric oxygen concentration (δ) in La1.5Sr0.5NiO4+δ , we found δ = −0.017, which corresponds to a doping level of n h = x + 2δ = 0.466. Also, a strong increase of the magnetization in the M(T) curve at temperatures below the spin-ordering temperature (T SO ~ 100 K) was observed. The M(H) curves show very small magnetic moments, which proves the weak ferromagnetic nature of La1.5Sr0.5NiO4+δ . The dependences of the dielectric constant on the frequency and the temperature, ε(ω, T) = ε′(ω, T) + iε″(ω, T), was investigated in the frequency range of 1–13 MHz. At temperatures around room temperature, the maximum of the real part (ε′) was higher than 105. Particularly, an abnormal dependence of the permittivity on frequency was observed. Depending on temperature, a dielectric resonance was observed at about 500 kHz or 8 MHz. Interestingly, we observed the dielectric-resonance effect with a negative permittivity. Such a feature is very similar to that observed in left-handed materials. The fitting of the experimental data for the dielectric constant at frequencies around the resonance frequency to the equations associated with an equivalent RLC series circuit proves that La1.5Sr0.5NiO4+δ belongs to the class of multiferroic materials.