A valve having a valve needle axially displaceable in a bore of a valve body and which has a conical valve needle tip having a radial valve sealing face, which in the closing position comes to rest on a conical inner wall of the valve body on the closed end of the bore. In the inner wall of the valve body there is at least one opening which connects the bore with the outside of the valve body. The valve includes two radial recesses disposed in the valve needle tip or in the inner wall of the valve body, and one of the radial recesses is located upstream and the other radial recess is located downstream from the position of the valve sealing face, in the closing position of the valve.