L(s) = 1 | + (0.866 + 0.5i)3-s + (0.499 + 0.866i)9-s + (−0.866 − 0.5i)11-s + 1.73i·17-s + 1.73·19-s + (−0.5 + 0.866i)25-s + 0.999i·27-s + (−0.499 − 0.866i)33-s + (1.5 − 0.866i)41-s + (0.866 − 1.5i)43-s + (0.5 + 0.866i)49-s + (−0.866 + 1.49i)51-s + (1.49 + 0.866i)57-s + (−0.866 + 0.5i)59-s + (−0.866 − 1.5i)67-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)3-s + (0.499 + 0.866i)9-s + (−0.866 − 0.5i)11-s + 1.73i·17-s + 1.73·19-s + (−0.5 + 0.866i)25-s + 0.999i·27-s + (−0.499 − 0.866i)33-s + (1.5 − 0.866i)41-s + (0.866 − 1.5i)43-s + (0.5 + 0.866i)49-s + (−0.866 + 1.49i)51-s + (1.49 + 0.866i)57-s + (−0.866 + 0.5i)59-s + (−0.866 − 1.5i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.581219923\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.581219923\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 - 1.73iT - T^{2} \) |
| 19 | \( 1 - 1.73T + T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + T^{2} \) |
| 41 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.154688586285954702486896848585, −8.669058049008190225213103231290, −7.58165257520905781028171427731, −7.52971498873554733325392915839, −5.93628350685361829433521148441, −5.42234042293614200634913079913, −4.30473027231193613378276981493, −3.52846938183535540873319029669, −2.74519341441249821020378530144, −1.58716519457094324597051653482,
1.09423754706444772878745002241, 2.54680630319399815907321155881, 2.94559019079676320023976896808, 4.20864469198996953266398000256, 5.07310533472400892319098449523, 6.01627085572829035552942210182, 7.10519655280252093738312942153, 7.55755064534450236182802521801, 8.122757432120616565243897586158, 9.243512969274602165085785505877