As I suspected in my comment above, there is an analog of Jensen's 'ancient' result for the Hurwitz zeta function: For $a>1/2$,
$$
\zeta(s,a)=\frac{\pi}{2(s-1)}\int_{-\infty}^\infty\frac{(a-1/2+it)^{1-s}}{\cosh^2(\pi t)}\, dt.
$$
This is (23) on p. 92 of *Series Associated with Zeta and Related Functions*, by Srivastava and Choi, Kluwer, 2001. (Unfortunately, without proof, but that's for another MO question.) For $a=1$ this reduces to Jensen.

Now let $d$ odd, and $\chi$ a nontrivial character modulo $d$, and $\epsilon=\pm1=\chi(-1)$.

So,
$$
L(s,\chi)=d^{-s}\sum_{d/2<j<d}\chi(j)(\zeta(s,j/d)+\epsilon\zeta(s,1-j/d)).
$$
Using the identity
$$
\zeta(s,a)=\zeta(s,a+1)+a^{-s}
$$
we get
$$
L(s,\chi)=d^{-s}\sum_{d/2<j<d}\chi(j)(\zeta(s,j/d)+\epsilon\zeta(s,2-j/d))+\epsilon\sum_{d/2<j<d}\chi(j)(d-j)^{-s}.
$$
Now the relevant Hurwitz zeta parameter is $>1/2$, so
$$
L(s,\chi)=\frac{\pi\cdot d^{-s}}{2(s-1)}\int_{-\infty}^\infty\sum_{d/2<j<d}\chi(j)((j/d-1/2+it)^{1-s}+\epsilon (3/2-j/d+it))^{1-s})\cosh^{-2}(\pi t)\, dt
$$
$$
+\epsilon\sum_{d/2<j<d}\chi(j)(d-j)^{-s}
$$
The singualrity at $s=1$ is removable: by the orthogonality of characters the sum over $j$ wil have a zero at $s=1$.