Load balancer health checks: how granular should they be

Hey everyone, I'm wondering how you all set up health checks for your load balancers in the cloud. We're getting a new service ready and I'm trying to decide how detailed these checks should be. Should we just see if the web server is running? Or should we ping a specific API endpoint to make sure the application logic is working right? What's your approach to finding a good balance between being thorough and adding too much overhead? Does it change based on how important the service is?