The Criminal Court in Riyadh held the first session for the 11 suspects on Thursday, the Saudi state news agency said.
In response to a question about the Saudi prosecutor's demand for the suspects' death penalty, the OHCHR Spokeswoman Ravina Shamdasani said on Friday that the office calls for an independent investigation "with global involvement".
"The Saudi Public Prosecutor submitted formal requests to Turkish authorities to officially share evidence and information related to this case, including any recordings in the possession of Turkish authorities".
The American comedian used his second episode, released October 28, to criticize Crown Prince Mohammed bin Salman over the killing of writer Jamal Khashoggi and the Saudi-led coalition at war in Yemen.
"This is why a UN-led and independent investigation is needed into the murder".
A comedian said Saudi Arabia encouraged more people to watch his show "Patriot Act" by making Netflix remove an episode.
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Mr Khashoggi - a former regime insider - had written columns for the Washington Post critical of Saudi Arabia's de-facto ruler Crown Prince Mohammed bin Salman.
Saudi Arabia has consistently maintained that Crown Prince Mohammed was not aware of and did not order the killing.
No date has been set for the next hearing and it was unclear how long the trial would last.
Assaf replaced Adel al-Jubeir.Experts in Saudi politics said the move reflected a perception that Jubeir was tainted by having served as Riyadh's chief global defender during the Khashoggi affair. Turkish security officials have kept up a slow leak of videos, photographs and morbid details surrounding Khashoggi's slaying to pressure the kingdom, as the two US-allied countries vie for influence over the wider Middle East.
"This demand was consistent with Saudi law and so we have removed it in Saudi only", the spokesperson said, adding that the move "doesn't mean we agree with these laws". "Three men carried five different pieces of luggage that contained Jamal Khashoggi's body parts". But in November the Saudi attorney general ruled out any involvement by the young crown prince.
Turkish officials said it had sought to extradite the suspects to Turkey to stand trail, but Saudi Arabia turned down the requests.
The prosecutor's office said ten other suspects were still under investigation.