They also sought life sentences for three senior crew members as the trial of Captain Lee Joon-Seok and 14 Sewol crew wound up in the southern city of Gwangju.
Lee "escaped the ship without making any efforts to rescue passengers", the prosecution team said in its sentencing recommendation to the court.
"He made excuses and lied. He showed no repentance... And so we ask for the death sentence," the prosecution said.
The three others should receive life sentences, they said, while recommending prison terms of between 15 and 30 years for the remaining 11 crew who were tried on lesser charges.
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The Sewol was carrying 476 passengers and crew when it sank off the southern coast on April 16. Only 174 people were rescued.
Lee and most of his crew were among the first to climb into rescue boats.