First recorded by Billy J. Kramer & The Dakotas (UK B-side #1 1963).
Hit album version by The Beatles (1964).
From the wiki: “John Lennon wrote the song prior to the formation of The Beatles. In 1963, he gave the song to Billy J. Kramer of The Dakotas, another Liverpool band signed to Parlophone by George Martin. Later, Lennon was reportedly dissatisfied with the Dakotas’ arrangement of his song as well as its position as a B-side (to the UK #1 ‘Bad to Me’, also written by Lennon-McCartney), so The Beatles recorded their own version, releasing it in the US on The Beatles’ Second Album and in the UK on the EP Long Tall Sally. The Beatles’ recording features George Harrison playing a Rickenbacker 360/12 guitar, giving to the world for the first time the distinctive sound of the famous guitar. ‘I Call Your Name’ was to have been included in the movie soundtrack to A Hard Day’s Night but was rejected in favor of ‘You Can’t Do That’.”
The Beatles, “I Call Your Name” (1964):