The seven judges have until September 1 to rule in the case in which opposition leader Raila Odinga alleges computer hacking and fraud handed victory to incumbent President Uhuru Kenyatta.
On Monday the court permitted lawyers for Odinga's National Super Alliance (NASA) coalition to have access to the computer servers and electronic voter kits of the Independent Electoral and Boundaries Commission (IEBC) in order to prove their allegations that the results were fiddled.
"The commission has nothing to hide," said IEBC lawyer Paul Muite.
James Orengo, a senior NASA official and lawyer, told the court that the alleged irregularities "cannot be considered as minor transgressions" leaving the result in question and requiring the annulment and rerunning of the vote.
More From This Section
Another opposition lawyer, Otiende Amollo, said tally sheets had been doctored in "a consistent pattern, that is: increase the votes for Uhuru Kenyatta and decrease the votes for Raila Odinga."
Odinga challenged his last election defeat in 2013, but lost in court. In 2007 widespread and violent street protests followed the declaration of Odinga's loss and on Monday a rights group said police had once again used excessive force to crush opposition protests following this month's vote.
"People have a right to protest peacefully, and Kenyan authorities should urgently put a stop to police abuse and hold those responsible to account."
HRW said police shot some people dead, wounded others, raped women and stole property while putting down the protests in parts of the country following opposition allegations of rigging.
At least 12 people were killed in the west of the country, including the city of Kisumu, HRW said, and more than 100 people were injured, but it added the true figure "is most likely much higher" as victims and relatives are reluctant to speak out.