| L(s) = 1 | + 5-s + 0.267·7-s + 1.46·11-s − 5.46·13-s − 0.535·17-s − 2·19-s − 3.73·23-s + 25-s − 1.53·29-s − 2·31-s + 0.267·35-s + 10.3·37-s − 9.92·41-s − 4.53·43-s − 0.267·47-s − 6.92·49-s − 6·53-s + 1.46·55-s − 14.3·59-s − 8.46·61-s − 5.46·65-s + 6.26·67-s − 9.46·71-s + 6.92·73-s + 0.392·77-s + 15.4·79-s + 13.1·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.101·7-s + 0.441·11-s − 1.51·13-s − 0.129·17-s − 0.458·19-s − 0.778·23-s + 0.200·25-s − 0.285·29-s − 0.359·31-s + 0.0452·35-s + 1.70·37-s − 1.55·41-s − 0.691·43-s − 0.0390·47-s − 0.989·49-s − 0.824·53-s + 0.197·55-s − 1.87·59-s − 1.08·61-s − 0.677·65-s + 0.765·67-s − 1.12·71-s + 0.810·73-s + 0.0447·77-s + 1.73·79-s + 1.44·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 - 0.267T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 13 | \( 1 + 5.46T + 13T^{2} \) |
| 17 | \( 1 + 0.535T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 3.73T + 23T^{2} \) |
| 29 | \( 1 + 1.53T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 10.3T + 37T^{2} \) |
| 41 | \( 1 + 9.92T + 41T^{2} \) |
| 43 | \( 1 + 4.53T + 43T^{2} \) |
| 47 | \( 1 + 0.267T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 14.3T + 59T^{2} \) |
| 61 | \( 1 + 8.46T + 61T^{2} \) |
| 67 | \( 1 - 6.26T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 6.92T + 73T^{2} \) |
| 79 | \( 1 - 15.4T + 79T^{2} \) |
| 83 | \( 1 - 13.1T + 83T^{2} \) |
| 89 | \( 1 - 9.92T + 89T^{2} \) |
| 97 | \( 1 + 8.92T + 97T^{2} \) |
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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
−8.145693696437440781237211565717, −7.64312649560779704318168366924, −6.66183823242826621240456410309, −6.15204437947550932623140120046, −5.09107219963562564572480395396, −4.56204830371029114281456510320, −3.47907348552524546368203961946, −2.45119853347985243936707860729, −1.62957260170116153096483168890, 0,
1.62957260170116153096483168890, 2.45119853347985243936707860729, 3.47907348552524546368203961946, 4.56204830371029114281456510320, 5.09107219963562564572480395396, 6.15204437947550932623140120046, 6.66183823242826621240456410309, 7.64312649560779704318168366924, 8.145693696437440781237211565717
