| L(s) = 1 | + 128i·2-s − 1.63e4·4-s + (−1.42e5 + 1.01e5i)5-s + 4.86e5i·7-s − 2.09e6i·8-s + (−1.29e7 − 1.82e7i)10-s + 9.64e7·11-s + 1.31e8i·13-s − 6.22e7·14-s + 2.68e8·16-s − 2.10e9i·17-s − 1.39e9·19-s + (2.33e9 − 1.66e9i)20-s + 1.23e10i·22-s + 2.88e10i·23-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s + (−0.814 + 0.580i)5-s + 0.223i·7-s − 0.353i·8-s + (−0.410 − 0.575i)10-s + 1.49·11-s + 0.579i·13-s − 0.157·14-s + 0.250·16-s − 1.24i·17-s − 0.357·19-s + (0.407 − 0.290i)20-s + 1.05i·22-s + 1.76i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.4082626587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4082626587\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 128iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.42e5 - 1.01e5i)T \) |
| good | 7 | \( 1 - 4.86e5iT - 4.74e12T^{2} \) |
| 11 | \( 1 - 9.64e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.31e8iT - 5.11e16T^{2} \) |
| 17 | \( 1 + 2.10e9iT - 2.86e18T^{2} \) |
| 19 | \( 1 + 1.39e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.88e10iT - 2.66e20T^{2} \) |
| 29 | \( 1 + 1.53e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + 2.58e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 5.01e11iT - 3.33e23T^{2} \) |
| 41 | \( 1 + 7.71e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 6.99e11iT - 3.17e24T^{2} \) |
| 47 | \( 1 - 2.83e12iT - 1.20e25T^{2} \) |
| 53 | \( 1 + 8.81e12iT - 7.31e25T^{2} \) |
| 59 | \( 1 - 1.68e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.85e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.14e13iT - 2.46e27T^{2} \) |
| 71 | \( 1 + 1.32e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.62e14iT - 8.90e27T^{2} \) |
| 79 | \( 1 + 2.51e13T + 2.91e28T^{2} \) |
| 83 | \( 1 + 3.38e14iT - 6.11e28T^{2} \) |
| 89 | \( 1 + 6.77e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 8.02e13iT - 6.33e29T^{2} \) |
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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
−11.33387230914279059511629752666, −9.623316877708877813328325049671, −8.857582721074705586128696827198, −7.44362771038035545985964379118, −6.87487058858181963739066323012, −5.60135084785186803662281360022, −4.19995035447478850777447997584, −3.35841730622221625772370502744, −1.61811863110590096259759641099, −0.10250818483645186089178914974,
0.895873919747209937521652857241, 1.97043239052747281878486238211, 3.71158532088718989682195906201, 4.12619455534202919313789449956, 5.61952948482512784495879507334, 7.08485213616191259415379400538, 8.414460749482787030945061068577, 9.103803477799781692118475623969, 10.49598101924221150482807606398, 11.35002641957008844455015714295
