Have you tried creating a smart list and bringing in the filter "First Name: is empty"?
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This has been asked before, and the conclusion was there's no matching syntax AFAIK.
BUT... this search may well be doomed anyway. You can't match on more than 1000 items and a field like firstName typically has low cardinality: just searched one of my small instances and 10% of the leads (22K) have no First Name. Of course that does depend on the maturity/management of your instance.
Also, when posting in the future you should mention [a] that you're using the API, not the UI and [b] which API endpoint(s) you're using.