They don't make you log in. Use this form [1] and they'll blacklist your email address system-wide [2]. Be sure to include any spelling variations (e.g. an optional dot) that may be in your contacts' address books:

  Please add my email addresses to the Do Not Contact List:
  
  [email protected]
  [email protected]
  [email protected]
  
  Thank you,
  
  Andrew

I did this in April and haven't since received anything but a confirmation from their customer support.

[1]: https://help.linkedin.com/app/ask/path/dnca

[2]: https://help.linkedin.com/app/answers/detail/a_id/426/