I am wondering if there are some know results for the non-trivial roots at ${\rm Re}(s) = \frac{1}{2}$, even maybe a table of the first few roots with $t>0$. This sister function


$$
L\_4^\* (s,\chi\_4) = \prod\_{p} \Big(1 + \frac{\chi\_4(p)}{p^{s}}\Big)^{-1} = \left(1-\frac{1}{4^s}\right)
\frac{\zeta(2s)}{L\_4(s,\chi\_4)}
$$


seems easier to handle. I could not find any study or table regarding the zeros of
$L\_4(s,\chi\_4)$. The only interesting document I found is the following (below) but it focuses on computational aspects only all done in C. Looking for help with this.


<https://fredrikj.net/blog/2016/11/dirichlet-l-functions-in-arb/>



Zeros of Dirichlet function $L(s,\chi_4)$
There is no need to use the subscript $4$ on the $L$-function: just write
$L(s,\chi\_4)$ and $L^\*(s,\chi\_4)$. The first nontrivial zero in the upper half of the critical strip is $1/2 + it$ where
$t \approx 6.020948i$.


Data for $L(s,\chi\_4)$ is available on the LMFDB: see <https://www.lmfdb.org/L/1/2e2/4.3/r1/0/0>. You can find imaginary parts of the initial zeros in the upper and lower half-plane towards the bottom of the page (the zeros are symmetric since $\chi\_4$ is quadratic).


In PARI, the command lfunzeros(L,T) computes imaginary parts $t$ of critical zeros with "$L$" being some $L$-function data (datum?) and $0 \leq t \leq T$. Being lazy, let's take L to be $x^2+1$, whose $L$-function is the zeta-function of $\mathbf Q(i)$, namely $\zeta(s)L(s,\chi\_4)$. Typing lfunzeros(x^2+1,20), I get the answer



> 
> [6.0209489046975966549025115216120858689, 10.243770304166554552137757479109959025, 12.988098012312422507453109789562993765, 14.134725141734693790457251983562470271, 16.342607104587222194976861483456149939, 18.291993196123534838526004277590699428].
> 
> 
> 


Only 14.1347... belongs to $\zeta(s)$; the rest are zeros of $L(s,\chi\_4)$.


See <https://pari.math.u-bordeaux.fr/pub/pari/manuals/2.13.3/refcard-lfun.pdf> for a printout listing PARI -function commands and <https://pari.math.u-bordeaux.fr/dochtml/html-stable/_L_minusfunctions.html> for a similar webpage.


