| L(s) = 1 | − 7.21·2-s + (−16.6 + 21.2i)3-s − 11.9·4-s + (−104. − 104. i)5-s + (119. − 153. i)6-s + (−266. − 266. i)7-s + 547.·8-s + (−175. − 707. i)9-s + (755. + 755. i)10-s + (−623. + 623. i)11-s + (198. − 254. i)12-s − 3.52e3·13-s + (1.92e3 + 1.92e3i)14-s + (3.97e3 − 486. i)15-s − 3.18e3·16-s + (3.31e3 + 3.62e3i)17-s + ⋯ |
| L(s) = 1 | − 0.901·2-s + (−0.615 + 0.787i)3-s − 0.186·4-s + (−0.838 − 0.838i)5-s + (0.555 − 0.710i)6-s + (−0.777 − 0.777i)7-s + 1.07·8-s + (−0.241 − 0.970i)9-s + (0.755 + 0.755i)10-s + (−0.468 + 0.468i)11-s + (0.115 − 0.147i)12-s − 1.60·13-s + (0.701 + 0.701i)14-s + (1.17 − 0.144i)15-s − 0.778·16-s + (0.675 + 0.737i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.257041 + 0.0997467i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.257041 + 0.0997467i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (16.6 - 21.2i)T \) |
| 17 | \( 1 + (-3.31e3 - 3.62e3i)T \) |
| good | 2 | \( 1 + 7.21T + 64T^{2} \) |
| 5 | \( 1 + (104. + 104. i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + (266. + 266. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 + (623. - 623. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + 3.52e3T + 4.82e6T^{2} \) |
| 19 | \( 1 - 5.00e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-6.08e3 + 6.08e3i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + (895. + 895. i)T + 5.94e8iT^{2} \) |
| 31 | \( 1 + (-7.44e3 + 7.44e3i)T - 8.87e8iT^{2} \) |
| 37 | \( 1 + (3.19e4 - 3.19e4i)T - 2.56e9iT^{2} \) |
| 41 | \( 1 + (1.21e3 - 1.21e3i)T - 4.75e9iT^{2} \) |
| 43 | \( 1 - 2.89e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 1.59e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.64e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.58e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + (2.82e3 + 2.82e3i)T + 5.15e10iT^{2} \) |
| 67 | \( 1 - 4.72e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (-3.47e5 - 3.47e5i)T + 1.28e11iT^{2} \) |
| 73 | \( 1 + (5.14e5 - 5.14e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 + (5.21e5 + 5.21e5i)T + 2.43e11iT^{2} \) |
| 83 | \( 1 - 3.14e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 7.72e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-6.67e5 + 6.67e5i)T - 8.32e11iT^{2} \) |
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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
−14.68335345310417317029917225171, −12.91852195090901861735187176623, −12.00517184864014252417454124214, −10.28684764258082739909217576593, −9.886356293275575579862467557609, −8.475513447717160363189527149002, −7.22456618481499252632774817316, −5.03230001781088427679679053360, −3.97676254043635769679549361380, −0.59511804325734668946712592899,
0.36113165614662926023504088244, 2.74575789052195063399201045730, 5.21338970985066250314119230624, 7.02227830064581211312402907925, 7.75293700632625282415120548768, 9.286903814099302142670092803853, 10.56511601429535716923682997087, 11.70300815940859548795720992274, 12.73510570492847251675311999681, 14.04360171906578006141914161292
