L(s) = 1 | + (0.766 + 0.642i)2-s + (1.73 − 0.0705i)3-s + (0.173 + 0.984i)4-s + (−0.918 − 0.334i)5-s + (1.37 + 1.05i)6-s + (−0.173 + 0.984i)7-s + (−0.500 + 0.866i)8-s + (2.99 − 0.244i)9-s + (−0.488 − 0.846i)10-s + (3.68 − 1.34i)11-s + (0.369 + 1.69i)12-s + (0.336 − 0.282i)13-s + (−0.766 + 0.642i)14-s + (−1.61 − 0.513i)15-s + (−0.939 + 0.342i)16-s + (1.75 + 3.04i)17-s + ⋯ |
L(s) = 1 | + (0.541 + 0.454i)2-s + (0.999 − 0.0407i)3-s + (0.0868 + 0.492i)4-s + (−0.410 − 0.149i)5-s + (0.559 + 0.432i)6-s + (−0.0656 + 0.372i)7-s + (−0.176 + 0.306i)8-s + (0.996 − 0.0813i)9-s + (−0.154 − 0.267i)10-s + (1.11 − 0.404i)11-s + (0.106 + 0.488i)12-s + (0.0933 − 0.0783i)13-s + (−0.204 + 0.171i)14-s + (−0.416 − 0.132i)15-s + (−0.234 + 0.0855i)16-s + (0.425 + 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 - 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.19443 + 0.879196i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19443 + 0.879196i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 + (-1.73 + 0.0705i)T \) |
| 7 | \( 1 + (0.173 - 0.984i)T \) |
good | 5 | \( 1 + (0.918 + 0.334i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-3.68 + 1.34i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.336 + 0.282i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.75 - 3.04i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.72 - 6.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.806 + 4.57i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (3.40 + 2.85i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.888 + 5.04i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.10 + 7.11i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.827 + 0.694i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (10.9 - 3.98i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.138 + 0.784i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 8.30T + 53T^{2} \) |
| 59 | \( 1 + (-0.398 - 0.145i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.275 + 1.56i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.61 + 4.71i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.65 + 9.79i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.18 - 12.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.48 - 7.12i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-6.15 - 5.16i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.02 + 1.77i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (13.7 - 4.99i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.83723286803183525645933020473, −10.48453755323016398624698771024, −9.413895855634658979689883519489, −8.386783969650679796166023974791, −7.988159996027664123619988950290, −6.67594564967216857661209168401, −5.81532277990660959576675473289, −4.13328748138840933885555006549, −3.66375313102556280623245753925, −2.02643026437266168033626262021,
1.64743769086139097390075987842, 3.15614605164498392390201810201, 3.96364843387482496618162008053, 5.00695715921494533439486002361, 6.75295862537992297181002109442, 7.31831575269044909156208142498, 8.680942730414472592543646403511, 9.433138409042048131534008602815, 10.31379318762311989807880079802, 11.41816051997051784731245691065