| L(s) = 1 | + (13.5 + 7.79i)3-s + (22.3 − 12.9i)5-s + (203. + 276. i)7-s + (121.5 + 210. i)9-s + (311. − 540. i)11-s + 3.25e3i·13-s + 402.·15-s + (275. + 158. i)17-s + (5.19e3 − 3.00e3i)19-s + (587. + 5.31e3i)21-s + (−21.0 − 36.3i)23-s + (−7.47e3 + 1.29e4i)25-s + 3.78e3i·27-s − 2.42e4·29-s + (1.75e4 + 1.01e4i)31-s + ⋯ |
| L(s) = 1 | + (0.5 + 0.288i)3-s + (0.178 − 0.103i)5-s + (0.592 + 0.805i)7-s + (0.166 + 0.288i)9-s + (0.234 − 0.405i)11-s + 1.48i·13-s + 0.119·15-s + (0.0560 + 0.0323i)17-s + (0.757 − 0.437i)19-s + (0.0634 + 0.573i)21-s + (−0.00172 − 0.00299i)23-s + (−0.478 + 0.829i)25-s + 0.192i·27-s − 0.995·29-s + (0.589 + 0.340i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.234 - 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.635258180\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.635258180\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-13.5 - 7.79i)T \) |
| 7 | \( 1 + (-203. - 276. i)T \) |
| good | 5 | \( 1 + (-22.3 + 12.9i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-311. + 540. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 - 3.25e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-275. - 158. i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (-5.19e3 + 3.00e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (21.0 + 36.3i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + 2.42e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.75e4 - 1.01e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-8.68e3 - 1.50e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 1.00e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 6.78e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (5.70e4 - 3.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (4.96e4 - 8.59e4i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (8.71e4 + 5.03e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.01e5 - 1.16e5i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-2.05e4 + 3.55e4i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.00e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-4.79e5 - 2.76e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (5.24e3 + 9.07e3i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 - 7.12e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (1.64e5 - 9.52e4i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 1.07e6iT - 8.32e11T^{2} \) |
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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
−10.95174653513196896736519385773, −9.454000584563145991070724475278, −9.156429755947909988570049368806, −8.130221235952948663470523383681, −7.06607084382057301584790699986, −5.82881020799101870200691987967, −4.83961694156397915460658751066, −3.71675034225589000610216667697, −2.40864438999169875505293907959, −1.40036200107102288596634810680,
0.56834400667003363598018608274, 1.67863363255396315987898908343, 2.99651880469770366157546189116, 4.10478852837240992505160545180, 5.31734922049569308446041146896, 6.49931967226127243773274010256, 7.76034810965710405743523026435, 7.984227251666579970980082479870, 9.457426485255003375743131651779, 10.18868536913690613180557869024
