The center knob is called the shower diverter and is a variation of a fauctt. Inside there is a stem and seat which works the same as in the hot & Cold facuetts.
When the valve is open the water flows to the spout (path of least resistance). When the valve is closed the path to the spout is closed and the water must now rise up to the shower head.
While it is necessary to shut the water off to service the hot & cold valves, the center diverter can be removed for servicing without turning the water off at the source or zone valves. (Be sure both the hot and cold faucetts are off though)
First remove the handle. There is probably a plastic trim cap on the center of the handle. Pry that out with a small flat screwdriver and you will see a screw inside. Remove that screw and the handle should pull off. Often if the handle have not been removed in a long time they are difficult to get off. In the trade we use an inexpensive Handle puller tool which is like a miniature gear puller)
Next remove the Trim escutcheon tube. Most of thenm are held in by a fine thread and can be unscrewed. I prefer to use a vinyl strap wrench to prevent scratching the chrome finish, but it can be done by wrapping a rag around the escutcheon an using a pair of channel lock pliers.
Once you have both the handle and trim escutcheon off you will see the diverter stem. (Ususally a brass body.)
Note where the handle shaft enters the diverter body. There is a packing nut there. Do not remove that nut. Under that nut there is a fiber packing that seals the shaft to prevent leaks from the shaft.)
There are normally some flats molded on the stem like a big nut to allow using a wrench to tighten or remove the diverter stem. If you can access those flats use a common adustable wrench to unscrew the diverter stem and pull it out of the mixer body.
If your mixer is recessed deep in the wall you may not be able to reach the flats with an ordinary wrench. If not, you will need a set of Plumbers sockets. (A full set of plumbers sockets can be bought at any hardware store for about $10.)
once you have the diverter stem removed look on the interior end of the shaft and you will see a common rubber faucett washer held in place by a Monel metal screw. Simple pull the screw and pry the old rubber bibb washer off and replace it with a new washer of the same sixe.
While you have the diverter out, check the faucett seat. It is inside the mixer body where the rubber washer makes contact. Often you will note a small groove on the lower part of the seat. If so, you will need to replace the seat also. ( I recommend always replacing the seat. They are cheap and once you have the stem out there easy to replace.)
The inside of the valve seat is machined with flats to allow installation and removal with a faucett seat wrench. The common seat wrench is a simple L shaped peice of metal that is tapered on each end. Some are tapered with 4 sides on one end and hexagonal on the other end. You insert the end of the seat wrench into the middle of the seat firmly, then unscrew the seat.
When installing the new seat, put a little bit of PTFE (Teflon) tape on the threads before installing. The easiest way to install is press the seat on the end of your seat wrench, then use the wrench to reach it into place and tighten it snug.
Before inserting the diverter stem, make sure you turn the shaft to the full open position to back the washer away from the seating area. then screw the diverter into the mixer body, tighten it in place with a wrencn and replace the trim escutcheon and handle.
I would expect the washer would cost about $.20 and a new seat about $1 or less, depending upon what brand and where you buy it. If you need to buy the handle puller or plumber sockets they are about $10 each and the seat wrench is about $5. Worst case scenario it will be under $30 in all, and you will have the tools for future projects.
The average time for an experienced plumber or maintenance tech is under 15 min. and i would expect a first time DIY'er should have no problem completing this in under 30 minutes.
Quite oftem people forget to unscrew the shaft when installing faucett washers. In that case, the new washer is forced into exactly the same position as the defective washer and it instantly distorts the new washer, which results in a leak.