| L(s) = 1 | + (−1.04 + 0.947i)2-s + (0.766 + 0.642i)3-s + (0.204 − 1.98i)4-s + (1.27 − 0.462i)5-s + (−1.41 + 0.0508i)6-s + (−3.79 + 2.18i)7-s + (1.66 + 2.28i)8-s + (0.173 + 0.984i)9-s + (−0.897 + 1.69i)10-s + (5.13 + 2.96i)11-s + (1.43 − 1.39i)12-s + (1.69 + 2.02i)13-s + (1.90 − 5.89i)14-s + (1.27 + 0.462i)15-s + (−3.91 − 0.815i)16-s + (0.539 − 3.05i)17-s + ⋯ |
| L(s) = 1 | + (−0.742 + 0.669i)2-s + (0.442 + 0.371i)3-s + (0.102 − 0.994i)4-s + (0.568 − 0.207i)5-s + (−0.576 + 0.0207i)6-s + (−1.43 + 0.827i)7-s + (0.590 + 0.807i)8-s + (0.0578 + 0.328i)9-s + (−0.283 + 0.534i)10-s + (1.54 + 0.893i)11-s + (0.414 − 0.401i)12-s + (0.470 + 0.560i)13-s + (0.509 − 1.57i)14-s + (0.328 + 0.119i)15-s + (−0.978 − 0.203i)16-s + (0.130 − 0.741i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0186 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 228 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0186 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.696788 + 0.709931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.696788 + 0.709931i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.04 - 0.947i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (-2.06 - 3.83i)T \) |
| good | 5 | \( 1 + (-1.27 + 0.462i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (3.79 - 2.18i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-5.13 - 2.96i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.69 - 2.02i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.539 + 3.05i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.866 - 2.38i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.70 + 0.476i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (5.45 + 9.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 6.57iT - 37T^{2} \) |
| 41 | \( 1 + (-3.24 + 3.86i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.790 + 2.17i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (2.17 - 0.383i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (2.95 - 8.11i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.11 + 11.9i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-5.27 - 1.91i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.626 + 3.55i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-2.55 + 0.928i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.12 - 3.46i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.29 + 2.76i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-5.76 + 3.32i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.11 - 3.71i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (11.9 + 2.10i)T + (91.1 + 33.1i)T^{2} \) |
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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
−12.45801938465408510551116447769, −11.40819489822698419675010558029, −9.774172395213829652007226403909, −9.519254408337916787645070848609, −8.964871909364854867274979189662, −7.45239012860853065916794244862, −6.39461541062666842323067664204, −5.59612726429829922867564990774, −3.86954091574799398172220287575, −1.98100392032488195987395453103,
1.11889088516671855719927230367, 3.04174163394892920801828557040, 3.80080593397189573122267337451, 6.34930557757524897026002571885, 6.86183053129892977999045927783, 8.307101045696072470407881249404, 9.178178334019666593161524741480, 9.977681899915983310985886037002, 10.80284221767159521982318478686, 11.95793086638454277761521821553
