L(s) = 1 | + (−0.346 − 0.938i)2-s + (−0.760 + 0.649i)4-s + (0.390 + 0.920i)5-s + (0.288 + 0.957i)7-s + (0.872 + 0.488i)8-s + (0.728 − 0.685i)10-s + (−0.746 − 0.665i)11-s + (0.568 + 0.822i)13-s + (0.798 − 0.601i)14-s + (0.155 − 0.987i)16-s + (−0.755 + 0.654i)17-s + (−0.983 + 0.182i)19-s + (−0.894 − 0.446i)20-s + (−0.365 + 0.930i)22-s + (−0.694 + 0.719i)25-s + (0.574 − 0.818i)26-s + ⋯ |
L(s) = 1 | + (−0.346 − 0.938i)2-s + (−0.760 + 0.649i)4-s + (0.390 + 0.920i)5-s + (0.288 + 0.957i)7-s + (0.872 + 0.488i)8-s + (0.728 − 0.685i)10-s + (−0.746 − 0.665i)11-s + (0.568 + 0.822i)13-s + (0.798 − 0.601i)14-s + (0.155 − 0.987i)16-s + (−0.755 + 0.654i)17-s + (−0.983 + 0.182i)19-s + (−0.894 − 0.446i)20-s + (−0.365 + 0.930i)22-s + (−0.694 + 0.719i)25-s + (0.574 − 0.818i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01636427874 + 0.3031020731i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01636427874 + 0.3031020731i\) |
\(L(1)\) |
\(\approx\) |
\(0.7492034705 + 0.006619369185i\) |
\(L(1)\) |
\(\approx\) |
\(0.7492034705 + 0.006619369185i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.346 - 0.938i)T \) |
| 5 | \( 1 + (0.390 + 0.920i)T \) |
| 7 | \( 1 + (0.288 + 0.957i)T \) |
| 11 | \( 1 + (-0.746 - 0.665i)T \) |
| 13 | \( 1 + (0.568 + 0.822i)T \) |
| 17 | \( 1 + (-0.755 + 0.654i)T \) |
| 19 | \( 1 + (-0.983 + 0.182i)T \) |
| 31 | \( 1 + (0.773 - 0.634i)T \) |
| 37 | \( 1 + (0.940 - 0.339i)T \) |
| 41 | \( 1 + (0.998 + 0.0475i)T \) |
| 43 | \( 1 + (-0.773 - 0.634i)T \) |
| 47 | \( 1 + (-0.294 + 0.955i)T \) |
| 53 | \( 1 + (-0.557 - 0.830i)T \) |
| 59 | \( 1 + (-0.928 + 0.371i)T \) |
| 61 | \( 1 + (-0.135 + 0.990i)T \) |
| 67 | \( 1 + (-0.665 - 0.746i)T \) |
| 71 | \( 1 + (0.794 - 0.607i)T \) |
| 73 | \( 1 + (-0.396 - 0.917i)T \) |
| 79 | \( 1 + (0.628 + 0.777i)T \) |
| 83 | \( 1 + (-0.810 - 0.585i)T \) |
| 89 | \( 1 + (-0.670 - 0.742i)T \) |
| 97 | \( 1 + (0.980 + 0.195i)T \) |
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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
−17.366415626319803574944732201529, −16.8117953923358285368394728030, −16.101573286695744517694571444172, −15.579349367664694660637016548647, −14.956786163133875111373565754767, −14.08713615168439550531740546184, −13.464295928263361458803092383489, −13.061394390511410541794893638456, −12.46317606717866180548282977062, −11.18532961145080565768031924325, −10.61465185295958056588817990849, −9.91438565670188302118385843447, −9.40744303309521835879374609804, −8.3778494837055361869824500380, −8.195801133832508711846543678555, −7.35147484739178680268080308169, −6.639580973934878663913947423629, −5.94437192278156989890705357564, −5.08023909948787641131481076785, −4.61856027789533186685592411904, −4.08253357404205701902674292945, −2.78122497271182046037682722939, −1.71097346702547723632965258155, −0.9698998375335990106239859252, −0.09406235944903323212444289771,
1.39730740969263833020555753783, 2.19759025112591413088655826625, 2.55135022343622633290337525546, 3.402171925556900554838867985379, 4.20017280993019382552498089269, 4.95481589169782957977770354664, 6.12590452241819174112212319208, 6.23168785774987570547518347604, 7.53667168136062839092318064173, 8.18674968042107241729792961137, 8.82567419368506153515963940701, 9.38590661499572767179289993876, 10.2313834914600314587754496259, 10.90950717425261316553026664394, 11.21237571210736072256879091758, 11.911090489815512615574067212130, 12.79924735947954947288895207567, 13.31612503957099723647741021166, 13.96784582183592113070638738562, 14.68925916431401017838844440366, 15.33236732541554803788133978326, 16.12708679570607038356493118719, 16.94271265522221473359079322995, 17.631609203245702576528457407938, 18.271630896756760379978760346558