L(s) = 1 | + (−0.987 + 0.156i)2-s + (−0.987 + 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (−0.760 + 0.649i)7-s + (−0.891 + 0.453i)8-s + (0.951 − 0.309i)9-s + (−0.972 − 0.233i)11-s + (−0.891 + 0.453i)12-s + (−0.760 − 0.649i)13-s + (0.649 − 0.760i)14-s + (0.809 − 0.587i)16-s + (−0.382 + 0.923i)17-s + (−0.891 + 0.453i)18-s + (−0.996 + 0.0784i)19-s + ⋯ |
L(s) = 1 | + (−0.987 + 0.156i)2-s + (−0.987 + 0.156i)3-s + (0.951 − 0.309i)4-s + (0.951 − 0.309i)6-s + (−0.760 + 0.649i)7-s + (−0.891 + 0.453i)8-s + (0.951 − 0.309i)9-s + (−0.972 − 0.233i)11-s + (−0.891 + 0.453i)12-s + (−0.760 − 0.649i)13-s + (0.649 − 0.760i)14-s + (0.809 − 0.587i)16-s + (−0.382 + 0.923i)17-s + (−0.891 + 0.453i)18-s + (−0.996 + 0.0784i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.418 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1011672870 - 0.06479136619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1011672870 - 0.06479136619i\) |
\(L(1)\) |
\(\approx\) |
\(0.3588293476 + 0.07560175869i\) |
\(L(1)\) |
\(\approx\) |
\(0.3588293476 + 0.07560175869i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 241 | \( 1 \) |
good | 2 | \( 1 + (-0.987 + 0.156i)T \) |
| 3 | \( 1 + (-0.987 + 0.156i)T \) |
| 7 | \( 1 + (-0.760 + 0.649i)T \) |
| 11 | \( 1 + (-0.972 - 0.233i)T \) |
| 13 | \( 1 + (-0.760 - 0.649i)T \) |
| 17 | \( 1 + (-0.382 + 0.923i)T \) |
| 19 | \( 1 + (-0.996 + 0.0784i)T \) |
| 23 | \( 1 + (0.382 + 0.923i)T \) |
| 29 | \( 1 + (0.707 - 0.707i)T \) |
| 31 | \( 1 + (0.923 + 0.382i)T \) |
| 37 | \( 1 + (-0.382 - 0.923i)T \) |
| 41 | \( 1 + (0.156 + 0.987i)T \) |
| 43 | \( 1 + (0.649 - 0.760i)T \) |
| 47 | \( 1 + (-0.707 - 0.707i)T \) |
| 53 | \( 1 + (0.707 - 0.707i)T \) |
| 59 | \( 1 + (0.707 + 0.707i)T \) |
| 61 | \( 1 + (-0.707 + 0.707i)T \) |
| 67 | \( 1 + (-0.707 + 0.707i)T \) |
| 71 | \( 1 + (-0.923 - 0.382i)T \) |
| 73 | \( 1 + (-0.760 + 0.649i)T \) |
| 79 | \( 1 + (-0.891 + 0.453i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (0.522 + 0.852i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−17.66396536192988014515493713476, −17.27320907468247106651385746811, −16.63503102437556100636096484027, −16.06524806292178159686284500350, −15.65759879265088563840450015265, −14.75321510991450963220968560321, −13.71661179816764249982187655409, −12.93287213960812471872161913917, −12.46175850971358882525898414412, −11.81291759242599142853833450122, −11.03631583387040085551493844304, −10.44719518332153419389085415923, −10.06229832534709618430982272771, −9.34077659571672906712366122489, −8.52753609300186534095873000043, −7.57754527171105429336479608552, −7.08568000486315237162285336423, −6.56311505780069985269177956926, −5.94083242401429004073067877204, −4.733868641493227477122877738091, −4.41457806322653053800164902271, −3.03236615387076871514022553105, −2.48330077900471361874440004376, −1.52935253233081154563351742532, −0.504508037252118669330773525204,
0.09914666130581527024307215019, 1.13735509468188054658512631520, 2.24106604902726405222943030231, 2.79005489020569010442476518713, 3.83408074661934302691698362352, 4.91648378306429332665814374555, 5.66812412338820369773866380119, 6.04626159603323017319414165155, 6.82151981933965532439275783407, 7.46639115296549587270232701494, 8.32894901784027458081356174607, 8.88907656835122228061757778070, 9.88490216744280227540542141742, 10.22171550349849119000226325681, 10.73042190776784490704257640358, 11.59366413291284493239196434527, 12.15913192204850279900232128067, 12.83641051404808504101185462610, 13.346636682742890751255221691049, 14.83800770120300831441802786789, 15.23067204973429758931759312537, 15.791819634173337304890713510798, 16.318510759959095270479539302516, 17.046431551426057764957768056930, 17.67740287136794974241106200767