| L(s) = 1 | + 0.618·2-s − 3-s − 1.61·4-s − 3.23·5-s − 0.618·6-s − 7-s − 2.23·8-s − 2·9-s − 2.00·10-s + 4.47·11-s + 1.61·12-s + 0.236·13-s − 0.618·14-s + 3.23·15-s + 1.85·16-s − 1.23·18-s − 7.23·19-s + 5.23·20-s + 21-s + 2.76·22-s − 23-s + 2.23·24-s + 5.47·25-s + 0.145·26-s + 5·27-s + 1.61·28-s − 1.47·29-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.577·3-s − 0.809·4-s − 1.44·5-s − 0.252·6-s − 0.377·7-s − 0.790·8-s − 0.666·9-s − 0.632·10-s + 1.34·11-s + 0.467·12-s + 0.0654·13-s − 0.165·14-s + 0.835·15-s + 0.463·16-s − 0.291·18-s − 1.66·19-s + 1.17·20-s + 0.218·21-s + 0.589·22-s − 0.208·23-s + 0.456·24-s + 1.09·25-s + 0.0286·26-s + 0.962·27-s + 0.305·28-s − 0.273·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + T + 3T^{2} \) |
| 5 | \( 1 + 3.23T + 5T^{2} \) |
| 11 | \( 1 - 4.47T + 11T^{2} \) |
| 13 | \( 1 - 0.236T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 7.23T + 19T^{2} \) |
| 29 | \( 1 + 1.47T + 29T^{2} \) |
| 31 | \( 1 + 9T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 + 2.23T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 + 3.47T + 47T^{2} \) |
| 53 | \( 1 - 11.2T + 53T^{2} \) |
| 59 | \( 1 + 1.52T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.70T + 73T^{2} \) |
| 79 | \( 1 + 7.23T + 79T^{2} \) |
| 83 | \( 1 - 6.47T + 83T^{2} \) |
| 89 | \( 1 - 8.94T + 89T^{2} \) |
| 97 | \( 1 - 3.70T + 97T^{2} \) |
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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
−12.21797440548279025324990261865, −11.70087467236819340133486872161, −10.64045259149951998424579849140, −9.064200128179016075487121723875, −8.413407376785878903737385167234, −6.90236653477683376803143602792, −5.74648271268377232924353020204, −4.33032702653995820299645518164, −3.57697056385459043134029716675, 0,
3.57697056385459043134029716675, 4.33032702653995820299645518164, 5.74648271268377232924353020204, 6.90236653477683376803143602792, 8.413407376785878903737385167234, 9.064200128179016075487121723875, 10.64045259149951998424579849140, 11.70087467236819340133486872161, 12.21797440548279025324990261865
