We study conformational transitions of a polymer on a simple-cubic lattice by calculating the zeros of the exact
partition function, up to chain length 24. In the complex temperature plane, two loci of the partition function
zeros are found for longer chains, suggesting the existence of both the coil-globule collapse transition and the
melting-freezing transition. The locus corresponding to coil-globule transition clearly approaches the real axis
as the chain length increases, and the transition temperature could be estimated by finite-size scaling. The form
of the logarithmic correction to the scaling of the partition function zeros could also be obtained. The other locus
does not show clear scaling behavior, but a supplementary analysis of the specific heat reveals a first-order-like
pseudotransition.