Intersection of Lines and Y-Intercept Determination

In the xy-plane, two lines, line $L_1$ and line $L_2$, intersect at point $P$. Line $L_1$ has a slope of $2$ and passes through the point $(1, 3)$. Line $L_2$ is perpendicular to line $L_1$ and intersects the y-axis at $b$. The equation of line $L_2$ can be expressed as $y = -\frac{1}{2}x + b$. If line $L_2$ passes through the point $P$, what is the value of $b$?