L(s) = 1 | + (0.439 + 0.761i)2-s + (−1.11 + 1.32i)3-s + (0.613 − 1.06i)4-s + 1.34·5-s + (−1.5 − 0.264i)6-s + 2.83·8-s + (−0.520 − 2.95i)9-s + (0.592 + 1.02i)10-s + 1.65·11-s + (0.726 + 1.99i)12-s + (1.68 + 2.91i)13-s + (−1.5 + 1.78i)15-s + (0.0209 + 0.0362i)16-s + (−0.233 − 0.405i)17-s + (2.02 − 1.69i)18-s + (1.61 − 2.79i)19-s + ⋯ |
L(s) = 1 | + (0.310 + 0.538i)2-s + (−0.642 + 0.766i)3-s + (0.306 − 0.531i)4-s + 0.602·5-s + (−0.612 − 0.107i)6-s + 1.00·8-s + (−0.173 − 0.984i)9-s + (0.187 + 0.324i)10-s + 0.498·11-s + (0.209 + 0.576i)12-s + (0.467 + 0.809i)13-s + (−0.387 + 0.461i)15-s + (0.00523 + 0.00906i)16-s + (−0.0567 − 0.0982i)17-s + (0.476 − 0.399i)18-s + (0.370 − 0.641i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.54687 + 0.766828i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.54687 + 0.766828i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.11 - 1.32i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.439 - 0.761i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 - 1.34T + 5T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + (-1.68 - 2.91i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.233 + 0.405i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.61 + 2.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + (3.13 - 5.42i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4.61 - 7.99i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.61 - 7.99i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.70 + 2.95i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.20 + 3.82i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.67 + 8.10i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.286 - 0.497i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.19 + 9.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.81 + 6.61i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (0.298 - 0.516i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.554T + 71T^{2} \) |
| 73 | \( 1 + (1.02 + 1.77i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.20 - 2.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.52 + 13.0i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.54 - 7.86i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.949 + 1.64i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.09418933271149358609166866000, −10.48549682056247329527793168971, −9.462992366450406661789618286963, −8.853756626599044319479436465524, −6.94858058933185393775940421850, −6.61834785531629225917214233497, −5.38230812697238564514380319898, −4.92223315931463880684090212539, −3.50526435110638626345219114257, −1.50792248904301098063192232970,
1.40000977447550774505957249211, 2.61316087231832447365979469722, 3.95165543361171249301758955541, 5.38436388529616352901938425534, 6.20360106806608713046304893021, 7.31625391447585698542672386689, 7.980938120989918386908790374740, 9.281272626305013244012971528598, 10.47206367295542940216042914310, 11.20483320511531534068090446810