Consider a triangle having vertices \( A(-2,3) \), \( B(1,9) \) and \( C(3,8) \). If a line \( L \) passing through the circum-center of triangle \( ABC \), bisects line \( BC \), and intersects y-axis at point \( \left(0,\dfrac{\alpha}{2}\right) \), then the value of real number \( \alpha \) is ________.
