| L(s) = 1 | + (−0.978 − 0.207i)2-s + (−1.72 − 0.152i)3-s + (0.913 + 0.406i)4-s + (1.62 + 1.53i)5-s + (1.65 + 0.507i)6-s + (−1.38 + 2.39i)7-s + (−0.809 − 0.587i)8-s + (2.95 + 0.524i)9-s + (−1.26 − 1.84i)10-s + (−2.02 − 0.430i)11-s + (−1.51 − 0.840i)12-s + (0.737 − 0.156i)13-s + (1.84 − 2.05i)14-s + (−2.56 − 2.90i)15-s + (0.669 + 0.743i)16-s + (−3.69 − 2.68i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (−0.996 − 0.0878i)3-s + (0.456 + 0.203i)4-s + (0.726 + 0.687i)5-s + (0.676 + 0.207i)6-s + (−0.522 + 0.904i)7-s + (−0.286 − 0.207i)8-s + (0.984 + 0.174i)9-s + (−0.401 − 0.582i)10-s + (−0.610 − 0.129i)11-s + (−0.437 − 0.242i)12-s + (0.204 − 0.0434i)13-s + (0.494 − 0.548i)14-s + (−0.662 − 0.748i)15-s + (0.167 + 0.185i)16-s + (−0.896 − 0.651i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.726 - 0.686i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.726 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.149022 + 0.374765i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.149022 + 0.374765i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 3 | \( 1 + (1.72 + 0.152i)T \) |
| 5 | \( 1 + (-1.62 - 1.53i)T \) |
| good | 7 | \( 1 + (1.38 - 2.39i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.02 + 0.430i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (-0.737 + 0.156i)T + (11.8 - 5.28i)T^{2} \) |
| 17 | \( 1 + (3.69 + 2.68i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.0538 + 0.0390i)T + (5.87 + 18.0i)T^{2} \) |
| 23 | \( 1 + (3.69 - 4.10i)T + (-2.40 - 22.8i)T^{2} \) |
| 29 | \( 1 + (0.355 + 3.38i)T + (-28.3 + 6.02i)T^{2} \) |
| 31 | \( 1 + (0.723 - 6.88i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (0.533 - 1.64i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (5.68 - 1.20i)T + (37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (5.69 - 9.86i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (0.660 + 6.28i)T + (-45.9 + 9.77i)T^{2} \) |
| 53 | \( 1 + (6.08 - 4.41i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (6.35 - 1.35i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-7.30 - 1.55i)T + (55.7 + 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.310 + 2.95i)T + (-65.5 - 13.9i)T^{2} \) |
| 71 | \( 1 + (4.06 - 2.95i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (0.246 + 0.757i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.45 + 13.8i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-14.3 + 6.38i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (0.320 + 0.987i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.910 - 8.66i)T + (-94.8 + 20.1i)T^{2} \) |
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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
−11.34799048633071476482790649771, −10.45978336080042682545864435679, −9.813491071461254861693692238563, −8.976419217914762038356102748161, −7.68175464359767131216524759798, −6.62828037529504964868423974173, −6.03796327397742206895636160967, −5.03511265192301156756874014764, −3.12782116423440003989934236852, −1.89389314591596337144106618361,
0.33795074538070989492216056555, 1.90645348021019964045195989559, 4.03732979271817738373999986404, 5.16253014808690625755082638203, 6.19972758352355854711943119994, 6.85641550762783160276911349976, 8.038940189785194661954803962008, 9.097674682038103168347895592392, 10.11246534323955368096995963359, 10.43128803989238060898283686028
