| L(s) = 1 | − 0.181·2-s + 3-s − 1.96·4-s + 3.75·5-s − 0.181·6-s + 0.719·8-s + 9-s − 0.681·10-s + 0.260·11-s − 1.96·12-s − 5.44·13-s + 3.75·15-s + 3.80·16-s + 0.644·17-s − 0.181·18-s − 4.44·19-s − 7.38·20-s − 0.0473·22-s − 9.20·23-s + 0.719·24-s + 9.09·25-s + 0.988·26-s + 27-s + 4.61·29-s − 0.681·30-s − 1.77·31-s − 2.12·32-s + ⋯ |
| L(s) = 1 | − 0.128·2-s + 0.577·3-s − 0.983·4-s + 1.67·5-s − 0.0740·6-s + 0.254·8-s + 0.333·9-s − 0.215·10-s + 0.0786·11-s − 0.567·12-s − 1.51·13-s + 0.969·15-s + 0.950·16-s + 0.156·17-s − 0.0427·18-s − 1.01·19-s − 1.65·20-s − 0.0100·22-s − 1.91·23-s + 0.146·24-s + 1.81·25-s + 0.193·26-s + 0.192·27-s + 0.856·29-s − 0.124·30-s − 0.318·31-s − 0.376·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6027 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 0.181T + 2T^{2} \) |
| 5 | \( 1 - 3.75T + 5T^{2} \) |
| 11 | \( 1 - 0.260T + 11T^{2} \) |
| 13 | \( 1 + 5.44T + 13T^{2} \) |
| 17 | \( 1 - 0.644T + 17T^{2} \) |
| 19 | \( 1 + 4.44T + 19T^{2} \) |
| 23 | \( 1 + 9.20T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 + 5.73T + 37T^{2} \) |
| 43 | \( 1 + 1.73T + 43T^{2} \) |
| 47 | \( 1 + 0.476T + 47T^{2} \) |
| 53 | \( 1 + 2.26T + 53T^{2} \) |
| 59 | \( 1 - 10.3T + 59T^{2} \) |
| 61 | \( 1 + 12.3T + 61T^{2} \) |
| 67 | \( 1 + 7.53T + 67T^{2} \) |
| 71 | \( 1 + 6.69T + 71T^{2} \) |
| 73 | \( 1 + 14.7T + 73T^{2} \) |
| 79 | \( 1 + 6.26T + 79T^{2} \) |
| 83 | \( 1 - 8.81T + 83T^{2} \) |
| 89 | \( 1 + 5.73T + 89T^{2} \) |
| 97 | \( 1 + 2.67T + 97T^{2} \) |
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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
−7.88190752364527334982529212336, −7.05617166867619935639695357194, −6.18335550044992341293783192460, −5.58263276602992610437735113195, −4.78975577868016669196666148888, −4.21141105559871024438113802484, −3.06931683100195556410526687188, −2.20314042896357978202569328232, −1.56973289587647892906186430572, 0,
1.56973289587647892906186430572, 2.20314042896357978202569328232, 3.06931683100195556410526687188, 4.21141105559871024438113802484, 4.78975577868016669196666148888, 5.58263276602992610437735113195, 6.18335550044992341293783192460, 7.05617166867619935639695357194, 7.88190752364527334982529212336
