A two-dimensional leaky-wave antenna (2-D LWA) is one that uses a cylindrically propagating leaky wave to form a narrow beam, which is usually a pencil beam at broadside when a horizontal dipole source is used. When excited with a vertical dipole source, however, a null at broadside is instead produced, which might be beneficial for some applications. This peculiar feature is scarcely investigated in the literature and a preliminary analysis is here reported. Relations between the half-power beamwidth and the half-power nullwidth of the beams radiated by 2-D LWAs are found and demonstrated through numerical results.