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In real-time systems, correctness depends not only on functionality but also on timeliness. A great deal of scheduling theories has been developed for verification of the temporal correctness of jobs (softwares) in such systems. Among them, the EDZL (Earliest Deadline first until Zero-Laxity) scheduling algorithm has received a growing attention thanks to its effectiveness in multi-core real-time scheduling. However, the true potential of EDZL has not yet been fully exploited in its schedulability analysis, as the state-of-the-art EDZL analysis techniques involve considerable pessimism. In this paper, we propose a new EDZL multi-core schedulability test. We first introduce an interesting observation that suggests an insight towards pessimism reduction in the schedulability analysis of EDZL. We then incorporate it into a well-known existing EDF (Earliest Deadline First) schedulability test, resulting in a new EDZL schedulability test. We demonstrate that the proposed EDZL test not only has lower time-complexity than existing EDZL schedulability tests, but also significantly improves the schedulability of EDZL by up to 36:6% compared to the best existing EDZL schedulability tests.