A ball is thrown from the location \( (x_{0}, y_{0}) = (0, 0) \) of a horizontal playground with an initial speed \( v_{0} \) at an angle \( \theta_{0} \) from the \( +x \)-direction. The ball is to be hit by a stone, which is thrown at the same time from the location \( (x_{1}, y_{1}) = (L, 0) \). The stone is thrown at an angle \( (180^{\circ} - \theta_{1}) \) from the \( +x \)-direction with a suitable initial speed. For a fixed \( v_{0} \), when \( (\theta_{0}, \theta_{1}) = (45^{\circ}, 45^{\circ}) \), the stone hits the ball after time \( T_{1} \), and when \( (\theta_{0}, \theta_{1}) = (60^{\circ}, 30^{\circ}) \), it hits the ball after time \( T_{2} \). In such a case, \( \left(\frac{T_{1}}{T_{2}}\right)^{2} \) is ______.
