Rolling bearings are a vital component for the safe functioning of rotating machinery. Long Short-Term Memory (LSTM) is extensively employed for predicting remaining useful life (RUL), but the gate mechanism of LSTM fails when the input data contains missing values. Most LSTM-based imputation methods only use the auto-correlation of time series, leading to unreliable imputation results and low prediction accuracy. This article proposed a LSTM Network with Auto-Correlation and Cross-Correlation (LSTM-AC) prediction model, which includes an imputation unit with both auto-correlation and cross-correlation to fully utilize the time series information and improve the credibility of the imputation results, Thus, the accuracy of RUL predictions is enhanced. Finally, the experimental results demonstrate the superiority of the proposed model over other models.