| L(s) = 1 | + (−1.35 − 0.118i)2-s + (−0.144 − 0.0254i)4-s + (−1.16 − 1.91i)5-s + (0.495 + 0.346i)7-s + (2.82 + 0.756i)8-s + (1.34 + 2.72i)10-s + (0.461 − 1.26i)11-s + (−0.157 − 1.80i)13-s + (−0.630 − 0.529i)14-s + (−3.46 − 1.26i)16-s + (−0.596 + 0.159i)17-s + (−6.66 + 3.84i)19-s + (0.119 + 0.305i)20-s + (−0.775 + 1.66i)22-s + (−4.35 − 6.22i)23-s + ⋯ |
| L(s) = 1 | + (−0.958 − 0.0838i)2-s + (−0.0722 − 0.0127i)4-s + (−0.519 − 0.854i)5-s + (0.187 + 0.131i)7-s + (0.998 + 0.267i)8-s + (0.426 + 0.862i)10-s + (0.139 − 0.382i)11-s + (−0.0437 − 0.499i)13-s + (−0.168 − 0.141i)14-s + (−0.865 − 0.315i)16-s + (−0.144 + 0.0387i)17-s + (−1.52 + 0.882i)19-s + (0.0266 + 0.0683i)20-s + (−0.165 + 0.354i)22-s + (−0.908 − 1.29i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 + 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0161493 - 0.236603i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0161493 - 0.236603i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.16 + 1.91i)T \) |
| good | 2 | \( 1 + (1.35 + 0.118i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-0.495 - 0.346i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-0.461 + 1.26i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.157 + 1.80i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (0.596 - 0.159i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (6.66 - 3.84i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.35 + 6.22i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (4.15 - 3.48i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.651 + 3.69i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.29 + 4.82i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (7.43 - 8.86i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.466 + 1.00i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-2.99 + 4.27i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (0.231 - 0.231i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.76 - 3.55i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-1.61 + 0.141i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-8.27 - 4.77i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.09 - 7.80i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (5.86 + 6.98i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.973 + 11.1i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (3.58 + 6.20i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-9.46 + 4.41i)T + (62.3 - 74.3i)T^{2} \) |
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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
−10.61733288721070894298525076849, −9.880660212464473228769230939409, −8.631119765948276955441694861212, −8.484751378089331810715499026088, −7.51122205463870390984983688126, −6.05412572123506190695080101960, −4.83119658846720635640322858717, −3.90369306241228806417755706070, −1.83909054762557329476055462676, −0.21115317029464260549721417595,
1.98393283198562795238463464506, 3.74718817655746625040364406470, 4.69183466811700825705306813106, 6.40836391799649027485686206786, 7.25759403153854456211493531646, 8.016903308943595655190303895451, 8.957880908811204461956184551846, 9.828238685019397916370533886119, 10.68309961542159295837483328148, 11.35226542048724963533266429576
