| L(s) = 1 | + 0.756·2-s + 1.98·3-s − 1.42·4-s + 5-s + 1.50·6-s − 1.14·7-s − 2.59·8-s + 0.955·9-s + 0.756·10-s + 1.66·11-s − 2.83·12-s − 1.28·13-s − 0.865·14-s + 1.98·15-s + 0.892·16-s − 17-s + 0.723·18-s − 7.24·19-s − 1.42·20-s − 2.27·21-s + 1.25·22-s + 5.37·23-s − 5.15·24-s + 25-s − 0.973·26-s − 4.06·27-s + 1.63·28-s + ⋯ |
| L(s) = 1 | + 0.535·2-s + 1.14·3-s − 0.713·4-s + 0.447·5-s + 0.614·6-s − 0.432·7-s − 0.916·8-s + 0.318·9-s + 0.239·10-s + 0.500·11-s − 0.819·12-s − 0.356·13-s − 0.231·14-s + 0.513·15-s + 0.223·16-s − 0.242·17-s + 0.170·18-s − 1.66·19-s − 0.319·20-s − 0.496·21-s + 0.268·22-s + 1.11·23-s − 1.05·24-s + 0.200·25-s − 0.190·26-s − 0.782·27-s + 0.308·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 71 | \( 1 - T \) |
| good | 2 | \( 1 - 0.756T + 2T^{2} \) |
| 3 | \( 1 - 1.98T + 3T^{2} \) |
| 7 | \( 1 + 1.14T + 7T^{2} \) |
| 11 | \( 1 - 1.66T + 11T^{2} \) |
| 13 | \( 1 + 1.28T + 13T^{2} \) |
| 19 | \( 1 + 7.24T + 19T^{2} \) |
| 23 | \( 1 - 5.37T + 23T^{2} \) |
| 29 | \( 1 - 3.00T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 - 5.92T + 37T^{2} \) |
| 41 | \( 1 - 3.01T + 41T^{2} \) |
| 43 | \( 1 + 4.58T + 43T^{2} \) |
| 47 | \( 1 + 4.13T + 47T^{2} \) |
| 53 | \( 1 + 5.50T + 53T^{2} \) |
| 59 | \( 1 + 7.71T + 59T^{2} \) |
| 61 | \( 1 - 5.64T + 61T^{2} \) |
| 67 | \( 1 + 5.94T + 67T^{2} \) |
| 73 | \( 1 + 12.9T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 9.16T + 83T^{2} \) |
| 89 | \( 1 + 1.57T + 89T^{2} \) |
| 97 | \( 1 + 7.48T + 97T^{2} \) |
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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
−7.970577806885039929999736755445, −6.88275888795791545643929072153, −6.30195846527812457090356026595, −5.52993514906499328689276490446, −4.56798693801934952775552306810, −4.10278338252439246324326536832, −3.10630606592522227028435544996, −2.70205353273119641342143971438, −1.57064391510069524418998674730, 0,
1.57064391510069524418998674730, 2.70205353273119641342143971438, 3.10630606592522227028435544996, 4.10278338252439246324326536832, 4.56798693801934952775552306810, 5.52993514906499328689276490446, 6.30195846527812457090356026595, 6.88275888795791545643929072153, 7.970577806885039929999736755445
