Large language models (LLMs), with advanced linguistic capabilities, have been employed in reranking tasks through a sequence-to-sequence approach. In this paradigm, multiple passages are reranked in a listwise manner and a textual reranked permutation is generated. However, due to the limited context window of LLMs, this reranking paradigm requires a sliding window strategy to iteratively handle larger candidate sets. This not only increases computational costs but also restricts the LLM from fully capturing all the comparison information for all candidates. To address these challenges, we propose a novel self-calibrated listwise reranking method, which aims to leverage LLMs to produce global relevance scores for ranking. To achieve it, we first propose the relevance-aware listwise reranking framework, which incorporates explicit list-view relevance scores to improve reranking efficiency and enable global comparison across the entire candidate set. Second, to ensure the comparability of the computed scores, we propose self-calibrated training that uses point-view relevance assessments generated internally by the LLM itself to calibrate the list-view relevance assessments. Extensive experiments and comprehensive analysis on the BEIR benchmark and TREC Deep Learning Tracks demonstrate the effectiveness and efficiency of our proposed method.