In the functional equation $L(s)$ is the Dirichlet Beta function which is defined as $L(s)= \sum_{n=0}^{\infty}\frac{\chi(n)}{n^s}$ where $\chi$ is a Dirichlet character of period 4.
Now I know that this $L(s)$ is an entire function of s, and all the terms on left of the functional equation are entire except for the $\Gamma(s)$ which has poles at $s=0,-1,-2, \ldots$.
For $s=-2n$ for $n \in \mathbb{N} \cup \{0\}$, $\sin\left(\frac{\pi}{2}\right)$ has simple zero and so there is not much of problem. I am having trouble understanding what happens when s is odd negative integer.
My guess is that $L$ function has zeros at negative odd integers. Any help is highly appreciated.