We present the results of a study of a particular carbon-chain-producing region, L1489 starless core (L1489 EMC), which is located at 1-arcmin east of L1489 IRS. We detected carbon-chain molecules (CCMs) HC_2_n+1N (n=1-3) and C_3_S in Ku band as well as high-energy excitation lines including C_4_H N=9-8, J=17/2-15/2, 19/2-17/2, and CH_3_CCH J=5-4, K=2 in the 3mm band toward a starless core called the eastern molecular core (EMC) of L1489 IRS. Maps of all the observed lines were also obtained. Comparisons with a number of early starless cores and WCCC source L1527 show that the column densities of C_4_H and CH_3_CCH are close to those of L1527, and the CH_3_CCH column densities of the EMC and L1527 are slightly higher than those of TMC-1. The EMC and L1527 have similar C3S column densities, but they are much lower than those of all the starless cores, with only 6.5% and 10% of the TMC-1 value, respectively. The emissions of the N-bearing species of the EMC and L1527 are at the medium level of the starless cores. These comparisons show that the CCM emissions in the EMC are similar to those of L1527, though L1527 contains a protostar. Although dark and quiescent, the EMC is warmer and at a later evolutionary stage than classical carbon-chain-producing regions in the cold, dark, quiescent early phase. The PACS, SPIRE, and SCUBA maps evidently show that the L1489 IRS seems to be the heating source of the EMC. Although it is located at the margins of the EMC, its bolometric luminosity and bolometric temperature are relatively high. Above all, the EMC is a rather particular carbon-chain-producing region and is quite significant for CCM science.