I feel like the real key is how much value are you providing in each interaction and what is in it for them? If you are only using the LI channel, I would suggest 3-4 touches over 4 weeks punched up with value.
That said, if there is a time-sensitive event or timeline, urgency to meet, you can punch it up or extend as necessary.
I typically perfect cycling channels (LI, Email, Phone, LI, Email, Phone).
If they are local, inviting to meet in person, or if you're doing a road show and they are a key account target, offering to meet at a local trade show, etc.
Let me know if this was helpful.
I have suggestions around tools depending on your tech stack to make this effective.