L(s) = 1 | + (−0.906 + 0.422i)2-s + (−1.85 + 2.64i)3-s + (0.642 − 0.766i)4-s + (0.560 − 3.17i)6-s + (3.57 − 0.958i)7-s + (−0.258 + 0.965i)8-s + (−2.53 − 6.96i)9-s + (−1.03 − 1.78i)11-s + (0.835 + 3.11i)12-s + (−1.78 + 1.25i)13-s + (−2.83 + 2.37i)14-s + (−0.173 − 0.984i)16-s + (0.543 + 1.16i)17-s + (5.24 + 5.24i)18-s + (1.10 + 4.21i)19-s + ⋯ |
L(s) = 1 | + (−0.640 + 0.298i)2-s + (−1.06 + 1.52i)3-s + (0.321 − 0.383i)4-s + (0.228 − 1.29i)6-s + (1.35 − 0.362i)7-s + (−0.0915 + 0.341i)8-s + (−0.844 − 2.32i)9-s + (−0.311 − 0.538i)11-s + (0.241 + 0.899i)12-s + (−0.496 + 0.347i)13-s + (−0.757 + 0.635i)14-s + (−0.0434 − 0.246i)16-s + (0.131 + 0.282i)17-s + (1.23 + 1.23i)18-s + (0.253 + 0.967i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.336839 + 0.726537i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.336839 + 0.726537i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.906 - 0.422i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-1.10 - 4.21i)T \) |
good | 3 | \( 1 + (1.85 - 2.64i)T + (-1.02 - 2.81i)T^{2} \) |
| 7 | \( 1 + (-3.57 + 0.958i)T + (6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (1.03 + 1.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.78 - 1.25i)T + (4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.543 - 1.16i)T + (-10.9 + 13.0i)T^{2} \) |
| 23 | \( 1 + (-3.68 + 0.322i)T + (22.6 - 3.99i)T^{2} \) |
| 29 | \( 1 + (-7.40 + 2.69i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-7.91 - 4.56i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.90 - 4.90i)T - 37iT^{2} \) |
| 41 | \( 1 + (7.67 - 1.35i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.866 - 9.90i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (8.82 + 4.11i)T + (30.2 + 36.0i)T^{2} \) |
| 53 | \( 1 + (-0.791 - 9.05i)T + (-52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-1.99 - 0.726i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-8.62 - 7.23i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.07 + 2.31i)T + (-43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (2.53 + 3.01i)T + (-12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-4.33 - 3.03i)T + (24.9 + 68.5i)T^{2} \) |
| 79 | \( 1 + (0.385 + 2.18i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (3.67 + 13.7i)T + (-71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (0.335 - 1.90i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-4.66 + 2.17i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.22693619796415897181094839276, −9.877436236541577422987373771442, −8.624031920973059304801646944433, −8.151683033603939780645977149062, −6.81684796507695887878829172587, −5.93897834423212988260631891294, −4.92941156572257183497882619258, −4.60570493540551771624831402155, −3.20667826804329381097585606949, −1.16229169427215548424229339441,
0.63913836264287681903067310821, 1.78832555332120080059821799575, 2.63207750326797259290698704362, 4.92765087388860524705796856192, 5.27427604628211123809416641825, 6.69093258904205018154827355308, 7.15492517257050066641864759011, 8.062735177740677471185907940263, 8.526337848889133607030520784209, 9.933313555824623942943140651640