1. A fuel injection controller for controlling a fuel injection amount of a fuel injector in an engine of a vehicle, the fuel injection controller comprising: an oxygen sensor that responds to an oxygen concentration inside an exhaust passage through which an exhaust of the engine passes; and an injection amount control unit programmed to control the fuel injection amount based on an output of the oxygen sensor, wherein the injection amount control unit includes: an injection amount correction value computing unit that determines an injection amount correction value based on the output of the oxygen sensor; a short-time learning value computing unit that determines, based on the injection amount correction value, a short-time learning value that is updated at a predetermined short-time learning speed; a long-time learning value computing unit that determines, based on the short-time learning value, a long-time learning value that is updated at a long-time learning speed that is slower than the short-time learning speed; a feedback correction amount computing unit that computes a feedback correction amount based on a sum of the injection amount correction value, the short-time learning value, and the long-time learning value; an injection amount control value computing unit that computes a control value of the fuel injection amount using the feedback correction amount; and a long-time learning value holding unit that stores the long-time learning value; wherein when the engine is started, the long-time learning value computing unit reads and uses a previous long-time learning value from the long-time learning value holding unit while the short-time learning value computing unit starts computing the short-time learning value anew without inheriting a previous short-time learning value.