L(s) = 1 | − 0.936i·3-s + 3.33i·5-s + 7-s + 2.12·9-s + 4.27i·11-s − 3.33i·13-s + 3.12·15-s + 2·17-s − 0.936i·19-s − 0.936i·21-s + 3.12·23-s − 6.12·25-s − 4.79i·27-s − 1.87i·29-s − 6.24·31-s + ⋯ |
L(s) = 1 | − 0.540i·3-s + 1.49i·5-s + 0.377·7-s + 0.707·9-s + 1.28i·11-s − 0.924i·13-s + 0.806·15-s + 0.485·17-s − 0.214i·19-s − 0.204i·21-s + 0.651·23-s − 1.22·25-s − 0.923i·27-s − 0.347i·29-s − 1.12·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.905 - 0.424i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.27618 + 0.283970i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.27618 + 0.283970i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
good | 3 | \( 1 + 0.936iT - 3T^{2} \) |
| 5 | \( 1 - 3.33iT - 5T^{2} \) |
| 11 | \( 1 - 4.27iT - 11T^{2} \) |
| 13 | \( 1 + 3.33iT - 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 0.936iT - 19T^{2} \) |
| 23 | \( 1 - 3.12T + 23T^{2} \) |
| 29 | \( 1 + 1.87iT - 29T^{2} \) |
| 31 | \( 1 + 6.24T + 31T^{2} \) |
| 37 | \( 1 - 1.87iT - 37T^{2} \) |
| 41 | \( 1 + 12.2T + 41T^{2} \) |
| 43 | \( 1 - 4.27iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 8.54iT - 53T^{2} \) |
| 59 | \( 1 + 7.60iT - 59T^{2} \) |
| 61 | \( 1 + 3.33iT - 61T^{2} \) |
| 67 | \( 1 + 15.7iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 - 9.47iT - 83T^{2} \) |
| 89 | \( 1 - 0.246T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−12.39039906459207917902234504372, −11.30621786416284103075021084271, −10.38048286852167911973342829444, −9.711772249668447394344779832936, −7.975565808397175921310096753890, −7.22715632871555151422167238199, −6.53894971324182652644403818233, −4.99683523001505722643861053315, −3.39907397124431242017305299152, −1.98277242281230573402704323001,
1.36393720080517626578335742800, 3.72924650051081452777375767085, 4.77508113317440899125380030023, 5.66068158236485121803569069640, 7.25857870779372428760683722162, 8.610602100665214182799760174926, 9.034482040763009840187007145919, 10.18412109204969958804887701260, 11.27269946673338102420603771211, 12.19585775275808448895316825889