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The ability to predict tandem mass (MS/MS) spectra from peptide sequences can significantly enhance our understanding of the peptide fragmentation process and could improve peptide identification in proteomics. However, current approaches for predicting high-energy collisional dissociation (HCD) spectra are limited to predict the intensities of expected ion types, i.e., the a/b/c/x/y/z ions and their neutral loss derivatives (referred to as backbone ions). In practice, backbone ions only account for < 70% of total ion intensities in HCD spectra, indicating many intense ions are ignored by current predictors. In this paper, we present a deep learning approach that can predict the complete spectra (both backbone and non-backbone ions) directly from peptide sequences. We made no assumptions or expectations on which kind of ions to predict but instead predicting the intensities for all possible m/z. Training this model needs no annotations of fragment ion nor any prior knowledge of the fragmentation rules. Our analyses show that the predicted 2+ and 3+ HCD spectra are highly similar to the experimental spectra, with average full-spectrum cosine similarities of 0.820 (±0.088) and 0.786 (±0.085), respectively, very close to the similarities between the experimental replicated spectra. In contrast, the best-performed backbone only models can only achieve an average similarity below 0.75 and 0.70 for 2+ and 3+ spectra, respectively. Furthermore, we developed a multi-task learning (MTL) approach for predicting spectra of insufficient training samples, which allows our model to make accurate predictions for electron transfer dissociation (ETD) spectra and HCD spectra of less abundant charges (1+ and 4+).