At the centre of Earth, the force due to upper half of the Earth will cancel the force due to lower half. In the similar manner, force due to any portion of the Earth at the centre will be cancelled due to the portion opposite to it. Thus, the gravitational force at the centre on any body will be $$0$$. Since, from Newton's law, we know $$F=mg$$. Since, mass $$m$$ of an object can never be $$0$$. Therefore, when $$F = 0$$, then $$g$$ has to be $$0$$. Thus, the value of $$g$$ is zero at the centre of Earth.