Advancements in deep neural networks have allowed automatic speech recognition (ASR) systems to attain human parity on several publicly available clean speech datasets. However, even state-of-the-art ASR systems experience performance degradation when confronted with adverse conditions, as a well-trained acoustic model is sensitive to variations in the speech domain, e.g., background noise. Intuitively, humans address this issue by relying on their linguistic knowledge: the meaning of ambiguous spoken terms is usually inferred from contextual cues thereby reducing the dependency on the auditory system. Inspired by this observation, we introduce the first open-source benchmark to utilize external large language models (LLMs) for ASR error correction, where N-best decoding hypotheses provide informative elements for true transcription prediction. This approach is a paradigm shift from the traditional language model rescoring strategy that can only select one candidate hypothesis as output transcription. The proposed benchmark contains a novel dataset, "HyPoradise" (HP), encompassing more than 316,000 pairs of N-best hypotheses and corresponding accurate transcriptions across prevalent speech domains. Given this dataset, we examine three types of error correction techniques based on LLMs with varying amounts of labeled hypotheses-transcription pairs, which gains significant word error rate (WER) reduction. Experimental evidence demonstrates the proposed technique achieves a breakthrough by surpassing the upper bound of traditional re-ranking based methods. More surprisingly, LLM with reasonable prompt design can even correct those tokens that are missing in N-best list. We make our results publicly accessible for reproducible pipelines with released pre-trained models, thus providing a new paradigm for ASR error correction with LLMs.