After a seven-year investigation and aborted settlement negotiations, UBS faces accusations of laundering the proceeds of tax fraud and illegally soliciting clients in France.
"Tax evasion is nothing more than theft against the community," prosecutor Eric Russo told the court.
Serge Roque, the trial's other prosecutor, added: "UBS has deliberately held, handled and managed accounts for people who were evading taxes."
Under French law, those convicted of money laundering can be ordered to pay a fine worth half the amount laundered. The prosecution estimates UBS's customers hid several billion euros from the French tax authorities.
On Wednesday, the lawyer representing the French government asked to be awarded 1.6 billion euros in damages.
UBS has denied wrongdoing. In a statement it said the trial's conclusions were "erroneous" and contested the amount of the fines.
"The PNF (National Financial Prosecutor's Office) has not provided details of the irrational way these amounts were calculated. The requested fine for UBS AG results from a simplistic approach taking into account the full amount paid by French taxpayers to normalise their situation," it said.
The prosecution also called for suspended jail sentences ranging from six months to 24 months for six UBS executives and former executives, as well as fines ranging from 50,000 euros to 500,000 euros.
(Reporting by Emmanuel Jarry and Inti Landauro; Editing by Richard Balmforth)
By Emmanuel Jarry and Inti Landauro