L(s) = 1 | + (−0.341 − 0.939i)2-s + (0.0448 + 0.998i)3-s + (−0.766 + 0.642i)4-s + (0.405 − 0.914i)5-s + (0.923 − 0.383i)6-s + (−0.503 + 0.863i)7-s + (0.865 + 0.500i)8-s + (−0.995 + 0.0896i)9-s + (−0.997 − 0.0689i)10-s + (−0.675 − 0.736i)12-s + (0.691 + 0.722i)13-s + (0.983 + 0.178i)14-s + (0.931 + 0.364i)15-s + (0.175 − 0.984i)16-s + (−0.982 + 0.185i)17-s + (0.424 + 0.905i)18-s + ⋯ |
L(s) = 1 | + (−0.341 − 0.939i)2-s + (0.0448 + 0.998i)3-s + (−0.766 + 0.642i)4-s + (0.405 − 0.914i)5-s + (0.923 − 0.383i)6-s + (−0.503 + 0.863i)7-s + (0.865 + 0.500i)8-s + (−0.995 + 0.0896i)9-s + (−0.997 − 0.0689i)10-s + (−0.675 − 0.736i)12-s + (0.691 + 0.722i)13-s + (0.983 + 0.178i)14-s + (0.931 + 0.364i)15-s + (0.175 − 0.984i)16-s + (−0.982 + 0.185i)17-s + (0.424 + 0.905i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9643132106 + 0.3287277998i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9643132106 + 0.3287277998i\) |
\(L(1)\) |
\(\approx\) |
\(0.7895779063 - 0.05327836549i\) |
\(L(1)\) |
\(\approx\) |
\(0.7895779063 - 0.05327836549i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (-0.341 - 0.939i)T \) |
| 3 | \( 1 + (0.0448 + 0.998i)T \) |
| 5 | \( 1 + (0.405 - 0.914i)T \) |
| 7 | \( 1 + (-0.503 + 0.863i)T \) |
| 13 | \( 1 + (0.691 + 0.722i)T \) |
| 17 | \( 1 + (-0.982 + 0.185i)T \) |
| 19 | \( 1 + (0.515 - 0.856i)T \) |
| 23 | \( 1 + (-0.770 + 0.636i)T \) |
| 29 | \( 1 + (0.573 + 0.819i)T \) |
| 31 | \( 1 + (-0.975 - 0.219i)T \) |
| 37 | \( 1 + (-0.161 - 0.986i)T \) |
| 41 | \( 1 + (-0.809 - 0.587i)T \) |
| 43 | \( 1 + (-0.289 - 0.957i)T \) |
| 47 | \( 1 + (0.644 + 0.764i)T \) |
| 53 | \( 1 + (0.634 - 0.773i)T \) |
| 59 | \( 1 + (-0.836 - 0.548i)T \) |
| 61 | \( 1 + (0.998 + 0.0552i)T \) |
| 67 | \( 1 + (0.675 + 0.736i)T \) |
| 71 | \( 1 + (0.783 - 0.620i)T \) |
| 73 | \( 1 + (0.0586 + 0.998i)T \) |
| 79 | \( 1 + (0.821 - 0.570i)T \) |
| 83 | \( 1 + (0.981 + 0.192i)T \) |
| 89 | \( 1 + (-0.256 + 0.966i)T \) |
| 97 | \( 1 + (0.996 + 0.0827i)T \) |
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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
−17.793271558387917613046981663893, −17.001998903243205062596267691329, −16.5431882575033791841798973514, −15.62767759773129028999583885528, −15.05735226401919540541937153252, −14.18161492246984412891806404587, −13.81825098746171426209577175769, −13.33212459684330237481391147953, −12.70431018925775783677964867433, −11.65543323055603260258002683129, −10.83223314976496301032193940742, −10.269554575963492218113016035776, −9.68581542374250338979978488355, −8.72996636560429365369737092942, −8.0475905913182521307873535297, −7.496184886819854276103202581752, −6.827460226898926677765214613030, −6.27479368015677296251659810641, −5.93712570014214466237935990289, −4.92735201641089085695416203081, −3.8316046157276410879833660266, −3.200227332594921910592470330921, −2.17118606768238671271130630886, −1.32718890058580704012735473635, −0.41897372644410233035421944745,
0.69345840983842664063111942147, 1.99816062720246638993270534724, 2.273974705000139538374590746377, 3.484173893871621138451789018813, 3.850332796919761654378210730417, 4.79299007372809964097618672591, 5.286641620109818358984454807982, 6.013309490824696109845485264876, 7.05044246158053774234045007684, 8.33581418673666706457845482161, 8.722738791060588455152281999598, 9.28658359658213669466037234696, 9.57344787985780071965466540509, 10.48479413645193651945794025886, 11.138449706525955518565354061742, 11.77105942805825329769190524299, 12.35684896476821188630599293680, 13.138580197238690615720226943230, 13.70855270241609490058185264942, 14.31078730376072791920988513352, 15.449904968357995396967929671486, 15.95632461781301023396242487125, 16.39630869807405324260898076798, 17.21832139912318319617614717106, 17.786502278994589166269031062458