A current $$i$$ is flowing in a straight conductor of length $$L$$. The magnetic induction at a point on its axis at a distance $$\frac{L}{4}$$ from its centre will be :

$$d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{i \, d\mathbf{l} \times \mathbf{\hat{r}}}{r^2}$$

$$|d\mathbf{l} \times \mathbf{\hat{r}}| = dl \sin\theta$$

For any point lying on the axis of a straight wire, the angle $$\theta$$ is either $$0^\circ$$ or $$180^\circ$$

$$\sin(0^\circ) = sin(180^\circ) = 0$$

Because the cross product is zero at every point on the conductor's axis, the total magnetic induction at any point on that axis (regardless of the distance from the center) is zero.