It is not typical for any e-mail client, or webmail client, to provide notifications unless it is running or you are logged in.
The PC (or Mac, for that matter) behave differently, by design, than a smartphone. You either need to have an email client running or be logged in to your Gmail (or Yahoo, or Outlook.com, or whatever webmail client) in a browser before notifications on new message arrival will be provided.
I guess it's possible that someone may have written an e-mail notifier that stays logged in to a given email account rather than expecting it to be open in the background like it typically is, but if one exists, I don't know about it.
Brian - Windows 10 Pro, 64-Bit, Version 20H2, Build 19042
If you think that you can think about a thing, inextricably attached to something else, without thinking of the thing it is attached to, then you have a legal mind.
~ Thomas Reed Powell