What is the PDF of this scaled non-central chi-squared random variable and what is the result of its integral?

Question

Consider a random variable that has a scaled non-central chi-squared distribution \begin{eqnarray*} L & = & a\chi_{1}^{2}(b^{2}), \end{eqnarray*} where $a$ is a positive scalar and $\chi_{1}^{2}(b^{2})$ represents a non-central chi-squared with one degree of freedom. In fact $\chi_{1}^{2}(b^{2})$ is the square of $\mathcal{N}(b,1)$. Denote the probability density function of $L$ by $p(s)$. What is the formula of $p(s)$ and what is the result of the following integral \begin{eqnarray*} I & = & \int_{h}^{\infty}p(s)ds \end{eqnarray*} please? Thanks.