L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.499 + 0.866i)4-s + (1.64 − 1.51i)5-s + 0.999i·6-s + (0.916 + 0.529i)7-s − 0.999·8-s + (0.499 + 0.866i)9-s + (2.13 + 0.664i)10-s − 0.825·11-s + (−0.866 + 0.499i)12-s + (−1.81 + 3.14i)13-s + 1.05i·14-s + (2.18 − 0.491i)15-s + (−0.5 − 0.866i)16-s + (0.742 + 1.28i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.499 + 0.288i)3-s + (−0.249 + 0.433i)4-s + (0.734 − 0.678i)5-s + 0.408i·6-s + (0.346 + 0.200i)7-s − 0.353·8-s + (0.166 + 0.288i)9-s + (0.675 + 0.210i)10-s − 0.248·11-s + (−0.249 + 0.144i)12-s + (−0.504 + 0.873i)13-s + 0.282i·14-s + (0.563 − 0.126i)15-s + (−0.125 − 0.216i)16-s + (0.179 + 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.621579254\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.621579254\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 5 | \( 1 + (-1.64 + 1.51i)T \) |
| 37 | \( 1 + (-5.95 - 1.24i)T \) |
good | 7 | \( 1 + (-0.916 - 0.529i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + 0.825T + 11T^{2} \) |
| 13 | \( 1 + (1.81 - 3.14i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.742 - 1.28i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.75 - 2.74i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 2.23T + 23T^{2} \) |
| 29 | \( 1 - 2.02iT - 29T^{2} \) |
| 31 | \( 1 - 6.27iT - 31T^{2} \) |
| 41 | \( 1 + (1.42 - 2.47i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 + 11.7iT - 47T^{2} \) |
| 53 | \( 1 + (-8.00 + 4.62i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.33 - 3.65i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.60 + 3.23i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (11.2 + 6.49i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.701 - 1.21i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 8.82iT - 73T^{2} \) |
| 79 | \( 1 + (3.10 + 1.79i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-13.4 + 7.74i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.86 - 1.65i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5.18T + 97T^{2} \) |
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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
−9.743091123774481247861144941506, −9.124164833416734124927864788988, −8.417710793480719175883220692049, −7.58878591370927189480221552424, −6.66002009342672735996576202292, −5.55752585681621551460057830034, −4.99882002941586491847652941477, −4.07312165640815351305672130580, −2.83049692581539653794391222826, −1.57888109698743060906846464971,
1.10002704318452504280456660775, 2.55413178794276525548994028529, 2.95474496127804779919179234645, 4.31042282936499068002592919564, 5.38473803215621397313139690344, 6.13019948054742800737349990299, 7.37622049276347482620179240738, 7.78747834857698267118489942467, 9.263264582840767576255180752103, 9.577448913579996822822249877293