How F5 LTM Handles Long Server Responses

What would happen to the connections below, especially connection 3 (open and active) when the server stops responding and Action on Service Down is set to None?

Connection 1 is brand new, still awaiting the response to the handshake SYN packet;  
Connection 2 is open but idle;
Connection 3 is open and active.

Let's say the keep alives are sent every 15 seconds so 46 seconds later the server is considered down.

I'm assuming you mean monitor probes, and not TCP Keepalives.

Connection 1 - brand new, still awaiting the response to the handshake SYN packet:

The LTM will retry SYNs until the server-side connection establishment fails.

Since you specify an http profile, the client-side is established and has issued a HTTP_REQUEST before the server-side starts the 3WHS - so the client gets a reset to close the Connection 1.

Connection 2 - open but idle

The pool member markdown does not impact existing, established connections.

The connection stays established until either the tcp idle timeout, or the client makes a request. This could be a TCP keepalive packet.

Once the client sends a packet destined for the pool member and fails to get a response on the server-side, there will be retries as defined by Max TCP retries, and if there is no response, resets will be sent to the client and pool member to forcibly close the connection.

Connection 3 - open and active

The pool member stops responding to packets from the LTM on the server-side connection.

Max TCP retries kicks in, and the LTM sends a reset to the pool member and the client to close out the connection.