L(s) = 1 | + (3.37 + 1.95i)5-s + 3.27i·7-s − 5.67·11-s + (−2.34 − 4.06i)13-s + (1.20 + 0.695i)17-s + (−3.88 − 1.97i)19-s + (−3.59 − 6.22i)23-s + (5.10 + 8.84i)25-s + (−4.52 + 2.61i)29-s + 9.36i·31-s + (−6.38 + 11.0i)35-s − 8.48·37-s + (3.32 + 1.91i)41-s + (−0.584 − 0.337i)43-s + (3.54 + 6.14i)47-s + ⋯ |
L(s) = 1 | + (1.51 + 0.872i)5-s + 1.23i·7-s − 1.71·11-s + (−0.651 − 1.12i)13-s + (0.292 + 0.168i)17-s + (−0.891 − 0.453i)19-s + (−0.749 − 1.29i)23-s + (1.02 + 1.76i)25-s + (−0.840 + 0.485i)29-s + 1.68i·31-s + (−1.07 + 1.87i)35-s − 1.39·37-s + (0.519 + 0.299i)41-s + (−0.0891 − 0.0514i)43-s + (0.517 + 0.896i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.986 - 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8941029446\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8941029446\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (3.88 + 1.97i)T \) |
good | 5 | \( 1 + (-3.37 - 1.95i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 - 3.27iT - 7T^{2} \) |
| 11 | \( 1 + 5.67T + 11T^{2} \) |
| 13 | \( 1 + (2.34 + 4.06i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.20 - 0.695i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.59 + 6.22i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (4.52 - 2.61i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 9.36iT - 31T^{2} \) |
| 37 | \( 1 + 8.48T + 37T^{2} \) |
| 41 | \( 1 + (-3.32 - 1.91i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.584 + 0.337i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.54 - 6.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.58 - 2.64i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.63 - 9.75i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.574 - 0.994i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.96 + 3.44i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.92 - 8.52i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.95 + 10.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.25 + 3.61i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.05T + 83T^{2} \) |
| 89 | \( 1 + (13.8 - 7.97i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.66 + 8.08i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.168903088581284482886393874626, −8.522765685334027746150597978218, −7.67852457287908345513678949525, −6.78269363152652059148915216232, −5.92908999691405860476378291489, −5.49087376810051440099202791705, −4.85546127946311504236497934625, −3.00929281498958056157651622389, −2.65264556356541152632420070254, −1.91616929238036067075587992273,
0.24472319049569255102427485485, 1.75856597407711000407466247947, 2.27607351405898066251183984678, 3.78915745194588702191218237737, 4.62963989780453205414983394833, 5.41032272798863272142470779668, 5.96160127065218785704469002679, 7.01587860953649882027232062600, 7.71109415395804575823361888767, 8.434348258220663860622695360061