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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/

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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 𝐿-function commands and https://pari.math.u-bordeaux.fr/dochtml/html-stable/_L_minusfunctions.html for a similar webpage.

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    $\begingroup$ Thank you, this answers my question. $\endgroup$
    – Vincent Granville
    Commented Jan 20, 2023 at 21:31
  • $\begingroup$ I wonder (too lazy to check just now) whether Sage would either have several of these examples in a look-up table, or at least be able to compute them (Riemann-Siegel and such?) quite quickly... $\endgroup$
    – paul garrett
    Commented Jan 20, 2023 at 21:57
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    $\begingroup$ @paulgarrett PARI-GP is in Sage, and in PARI the command lfunzeros(L,T) computes imaginary parts of critical zeros with $0 \leq t \leq T$. Being lazy, I took 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 got the answer [6.0209489046975966549025115216120858689, 10.243770304166554552137757479109959025, 12.988098012312422507453109789562993765, 14.134725141734693790457251983562470271, 16.342607104587222194976861483456149939, 18.291993196123534838526004277590699428]. Only 14.1347... belongs to $\zeta(s)$. $\endgroup$
    – KConrad
    Commented Jan 20, 2023 at 22:08
  • $\begingroup$ @KConrad, ah, thanks for checking! :) $\endgroup$
    – paul garrett
    Commented Jan 20, 2023 at 22:09
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    $\begingroup$ @sku you did not define $N(t)$ in your question. If it is counting the number of zeros of $L(s,\chi_4)$ in the critical strip with imaginary part having absolute value at most $t$, then an estimate for that is known of the same quality as for the Riemann zeta-function. See Chap. 16 of Davenport's Multiplicative Number Theory, Chap. 14 of Montgomery and Vaughan's Multiplicative Number Theory I, and Chap. 5 (Theorem 5.8) of Iwaniec-Kowalski's Analytic Number Theory. $\endgroup$
    – KConrad
    Commented Apr 24, 2023 at 2:01

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