The week of March 21, 1964, saw The Beatles dominating the top of the Billboard Hot 100. ‘She Loves You‘ and ‘I Want to Hold Your Hand‘ occupied the #1 and #2 spots, respectively, and a re-release of ‘Please Please Me‘ — which had bombed in the U.S. a year earlier — made it to #3.
In the #4 slot that week was ‘Dawn (Go Away)’ by The Four Seasons, a song that peaked at #3 during a six-week stint in the top ten. During that span, it was never topped by anything but a Beatles song.
‘Dawn (Go Away)’ was one of seven top five singles The Four Seasons had between 1962 and 1965. The song started as a folk tune before the band had it arranged to fit their signature pop sound.
They call her Dawn
Dawn
Go away I’m no good for you
Oh Dawn
Stay with him, he’ll be good to you
Hang on
Hang on to you
Think
What a big man he’ll be
Think
Of the places you’ll see
Now think what the future would be with a poor boy like me
Dawn go away
Please go away
Although I know
I want you to stay
Dawn go away
Please go away
Baby, don’t cry
It’s better this way
Ahh, ahh, ah
Ohh-ohh-oh
Dawn
Go away back where you belong
Girl we can’t
Change the places where we were born
Before you say
That you want me
I want you to think
What your family would say
Think
What your throwing away
Now think what the future would be with a poor boy like me
Meee-ee
Dawn
Go away I’m no good for you
Dawn
Go away I’m no good for you
Meet Me In Montauk 
I’m not sure if this song was in Jersey Boys, but I’m really not familiar with it like their other hits.