| L(s) = 1 | + (−0.142 − 0.989i)2-s + (−1.58 − 0.705i)3-s + (−0.959 + 0.281i)4-s + (−1.76 + 1.37i)5-s + (−0.473 + 1.66i)6-s + (−3.04 + 1.95i)7-s + (0.415 + 0.909i)8-s + (2.00 + 2.23i)9-s + (1.61 + 1.55i)10-s + (−0.00228 + 0.0159i)11-s + (1.71 + 0.231i)12-s + (1.50 − 2.34i)13-s + (2.36 + 2.73i)14-s + (3.76 − 0.925i)15-s + (0.841 − 0.540i)16-s + (1.07 − 3.66i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 − 0.699i)2-s + (−0.913 − 0.407i)3-s + (−0.479 + 0.140i)4-s + (−0.789 + 0.614i)5-s + (−0.193 + 0.680i)6-s + (−1.14 + 0.738i)7-s + (0.146 + 0.321i)8-s + (0.667 + 0.744i)9-s + (0.509 + 0.490i)10-s + (−0.000689 + 0.00479i)11-s + (0.495 + 0.0669i)12-s + (0.417 − 0.650i)13-s + (0.632 + 0.730i)14-s + (0.971 − 0.239i)15-s + (0.210 − 0.135i)16-s + (0.260 − 0.888i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0984 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0984 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.435644 - 0.394679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.435644 - 0.394679i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (1.58 + 0.705i)T \) |
| 5 | \( 1 + (1.76 - 1.37i)T \) |
| 23 | \( 1 + (3.58 + 3.18i)T \) |
| good | 7 | \( 1 + (3.04 - 1.95i)T + (2.90 - 6.36i)T^{2} \) |
| 11 | \( 1 + (0.00228 - 0.0159i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.50 + 2.34i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-1.07 + 3.66i)T + (-14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-0.620 - 2.11i)T + (-15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.90 + 6.50i)T + (-24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.30 - 5.04i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-7.30 - 8.43i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (3.69 + 3.19i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (0.521 - 1.14i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 8.02T + 47T^{2} \) |
| 53 | \( 1 + (2.54 + 3.95i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.07 + 1.66i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.98 + 2.73i)T + (39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.47 - 10.2i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-12.0 + 1.72i)T + (68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (4.01 + 13.6i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-4.69 + 7.30i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-0.587 + 0.509i)T + (11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-4.77 + 10.4i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-5.55 + 6.40i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.26050481812979585703669165123, −9.854236196679464363437279713261, −8.491193003006909855091740396592, −7.67442720997039758700561609468, −6.57507488781417247727628638356, −5.92059012650939472040946125758, −4.69042575804687830412149989511, −3.44912125932917314289606867064, −2.50679507891386421790606273378, −0.53266137407990876345505625900,
0.874298498571830399799488378584, 3.72384673129994947717562066900, 4.16235366456910621671069032310, 5.35571940048779261606862724448, 6.28762655720155356453931984686, 7.01067809091098786572827443874, 7.917955631534430738965908441916, 9.075386806155575483345262745969, 9.697661126647560949151772928796, 10.62343710260636824474974740560
