L(s) = 1 | + (1.59 + 1.16i)2-s + (0.896 + 2.76i)4-s + (−0.533 − 0.734i)5-s + (−1.16 + 3.57i)8-s − 1.79i·10-s + (−0.156 + 0.987i)11-s + (0.587 − 0.809i)13-s + (−3.65 + 2.65i)16-s + (1.54 − 2.13i)20-s + (−1.39 + 1.39i)22-s + (0.0542 − 0.166i)25-s + (1.87 − 0.610i)26-s − 5.17·32-s + (3.24 − 1.05i)40-s + (0.437 − 1.34i)41-s + ⋯ |
L(s) = 1 | + (1.59 + 1.16i)2-s + (0.896 + 2.76i)4-s + (−0.533 − 0.734i)5-s + (−1.16 + 3.57i)8-s − 1.79i·10-s + (−0.156 + 0.987i)11-s + (0.587 − 0.809i)13-s + (−3.65 + 2.65i)16-s + (1.54 − 2.13i)20-s + (−1.39 + 1.39i)22-s + (0.0542 − 0.166i)25-s + (1.87 − 0.610i)26-s − 5.17·32-s + (3.24 − 1.05i)40-s + (0.437 − 1.34i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.390 - 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.287894182\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.287894182\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (0.156 - 0.987i)T \) |
| 13 | \( 1 + (-0.587 + 0.809i)T \) |
good | 2 | \( 1 + (-1.59 - 1.16i)T + (0.309 + 0.951i)T^{2} \) |
| 5 | \( 1 + (0.533 + 0.734i)T + (-0.309 + 0.951i)T^{2} \) |
| 7 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 19 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 31 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
| 37 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 41 | \( 1 + (-0.437 + 1.34i)T + (-0.809 - 0.587i)T^{2} \) |
| 43 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (1.87 + 0.610i)T + (0.809 + 0.587i)T^{2} \) |
| 53 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 59 | \( 1 + (-0.297 + 0.0966i)T + (0.809 - 0.587i)T^{2} \) |
| 61 | \( 1 + (-0.951 - 1.30i)T + (-0.309 + 0.951i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (-0.831 - 1.14i)T + (-0.309 + 0.951i)T^{2} \) |
| 73 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 79 | \( 1 + (0.951 - 1.30i)T + (-0.309 - 0.951i)T^{2} \) |
| 83 | \( 1 + (0.734 - 0.533i)T + (0.309 - 0.951i)T^{2} \) |
| 89 | \( 1 + 1.41iT - T^{2} \) |
| 97 | \( 1 + (-0.309 - 0.951i)T^{2} \) |
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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
−10.18057014718303809377922886591, −8.687510706465072572562774882571, −8.326181386738464811678777577107, −7.36784406263628965364125873767, −6.84010561591414491351419134984, −5.67396018705017804628265135828, −5.16393092533994317645232428871, −4.24346851001654247700603295881, −3.62644016560264785890134462195, −2.35880500937058903685276192025,
1.41722910273184003910448332983, 2.78444417513778668619674776250, 3.39524180351918705897042324376, 4.22540105407334068008902256460, 5.11932604792723256848952119838, 6.22434846355825818535664234387, 6.56645607170902626369601225995, 7.83288390545035844758123526162, 9.156889809150253013478527459150, 9.964245369269890980073335957081