There is evidence of misinformation in the online discourses and discussions about the COVID-19 vaccines. Using a sample of 1.6 million geotagged English tweets and the data from the CDC COVID Data Tracker, we conduct a quantitative study to understand the influence of both misinformation and fact-based news on Twitter on the COVID-19 vaccine uptake in the U.S. from April 19 when U.S. adults were vaccine eligible to May 7, 2021, after controlling state-level factors such as demographics, education, and the pandemic severity. We identify the tweets related to either misinformation or fact-based news by analyzing the URLs. By analyzing the content of the most frequent tweets of these two groups, we find that their structures are similar, making it difficult for Twitter users to distinguish one from another by reading the text alone. The users who spread both fake news and fact-based news tend to show a negative attitude towards the vaccines. We further conduct the Fama-MacBeth regression with the Newey-West adjustment to examine the effect of fake-news-related and fact-related tweets on the vaccination rate, and find marginally negative correlations.