| L(s) = 1 | − 2.37·3-s + 0.438·5-s + 3.46·7-s + 2.62·9-s − 3.15·11-s − 5.09·13-s − 1.03·15-s − 2.92·17-s + 3.68·19-s − 8.22·21-s + 7.97·23-s − 4.80·25-s + 0.896·27-s + 1.62·29-s − 2.09·31-s + 7.48·33-s + 1.51·35-s − 5.18·37-s + 12.0·39-s − 5.08·41-s − 1.41·43-s + 1.14·45-s + 1.50·47-s + 5.01·49-s + 6.92·51-s − 9.99·53-s − 1.38·55-s + ⋯ |
| L(s) = 1 | − 1.36·3-s + 0.196·5-s + 1.31·7-s + 0.874·9-s − 0.951·11-s − 1.41·13-s − 0.268·15-s − 0.708·17-s + 0.845·19-s − 1.79·21-s + 1.66·23-s − 0.961·25-s + 0.172·27-s + 0.301·29-s − 0.376·31-s + 1.30·33-s + 0.256·35-s − 0.852·37-s + 1.93·39-s − 0.794·41-s − 0.216·43-s + 0.171·45-s + 0.219·47-s + 0.716·49-s + 0.969·51-s − 1.37·53-s − 0.186·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
| good | 3 | \( 1 + 2.37T + 3T^{2} \) |
| 5 | \( 1 - 0.438T + 5T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 11 | \( 1 + 3.15T + 11T^{2} \) |
| 13 | \( 1 + 5.09T + 13T^{2} \) |
| 17 | \( 1 + 2.92T + 17T^{2} \) |
| 19 | \( 1 - 3.68T + 19T^{2} \) |
| 23 | \( 1 - 7.97T + 23T^{2} \) |
| 29 | \( 1 - 1.62T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 + 5.18T + 37T^{2} \) |
| 41 | \( 1 + 5.08T + 41T^{2} \) |
| 43 | \( 1 + 1.41T + 43T^{2} \) |
| 47 | \( 1 - 1.50T + 47T^{2} \) |
| 53 | \( 1 + 9.99T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 14.9T + 67T^{2} \) |
| 71 | \( 1 - 0.264T + 71T^{2} \) |
| 73 | \( 1 - 14.9T + 73T^{2} \) |
| 79 | \( 1 + 12.5T + 79T^{2} \) |
| 83 | \( 1 + 6.13T + 83T^{2} \) |
| 89 | \( 1 + 14.1T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−9.365324889952775630194931249555, −8.276116605558874312734466874717, −7.44131699961883882062556934208, −6.78899372824005794903707457964, −5.54245953542059837806068338978, −5.10333066736730761700571819183, −4.56881823061230969529300013576, −2.81771297289894240158767617731, −1.56244122897092035302677151413, 0,
1.56244122897092035302677151413, 2.81771297289894240158767617731, 4.56881823061230969529300013576, 5.10333066736730761700571819183, 5.54245953542059837806068338978, 6.78899372824005794903707457964, 7.44131699961883882062556934208, 8.276116605558874312734466874717, 9.365324889952775630194931249555
