L(s) = 1 | + (0.809 − 0.587i)2-s + (−0.744 − 2.29i)3-s + (0.309 − 0.951i)4-s + (2.77 + 2.01i)5-s + (−1.94 − 1.41i)6-s + (0.326 − 1.00i)7-s + (−0.309 − 0.951i)8-s + (−2.26 + 1.64i)9-s + 3.43·10-s + (0.637 − 3.25i)11-s − 2.40·12-s + (0.266 − 0.193i)13-s + (−0.326 − 1.00i)14-s + (2.55 − 7.86i)15-s + (−0.809 − 0.587i)16-s + (2.82 + 2.04i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.429 − 1.32i)3-s + (0.154 − 0.475i)4-s + (1.24 + 0.902i)5-s + (−0.795 − 0.577i)6-s + (0.123 − 0.379i)7-s + (−0.109 − 0.336i)8-s + (−0.755 + 0.548i)9-s + 1.08·10-s + (0.192 − 0.981i)11-s − 0.695·12-s + (0.0738 − 0.0536i)13-s + (−0.0872 − 0.268i)14-s + (0.659 − 2.03i)15-s + (−0.202 − 0.146i)16-s + (0.684 + 0.496i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.215 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.24470 - 1.54931i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.24470 - 1.54931i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (-0.637 + 3.25i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
good | 3 | \( 1 + (0.744 + 2.29i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-2.77 - 2.01i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.326 + 1.00i)T + (-5.66 - 4.11i)T^{2} \) |
| 13 | \( 1 + (-0.266 + 0.193i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.82 - 2.04i)T + (5.25 + 16.1i)T^{2} \) |
| 23 | \( 1 + 7.90T + 23T^{2} \) |
| 29 | \( 1 + (1.91 - 5.89i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.78 - 1.29i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.98 + 6.10i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.26 + 3.88i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 4.47T + 43T^{2} \) |
| 47 | \( 1 + (-3.18 - 9.80i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.313 - 0.227i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.235 - 0.725i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-2.77 - 2.01i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 + (-10.1 - 7.40i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.34 - 4.14i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-12.2 + 8.90i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.76 - 5.64i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 6.50T + 89T^{2} \) |
| 97 | \( 1 + (-3.88 + 2.82i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.92742959263146034868690773560, −10.45989042420471464392432307055, −9.326092901421108940376833914515, −7.86918323889028852563716633881, −6.94395444864420684879488788856, −5.96407532094455546174503607862, −5.73784065853850524738322800127, −3.72077050195074842850593254621, −2.38587261832254410321862113557, −1.31637325951306153208624672463,
2.12527627548866563137543671478, 3.94172306940222660437012476315, 4.85388516341643636967688920305, 5.47683539137570688238402604212, 6.27841415375831148667659333120, 7.80308483755859113701003980007, 9.045021474412870504441091298421, 9.718924133935561965500019798598, 10.24726517690954173231490956730, 11.63675231517685402847186081739