L(s) = 1 | + 1.82·2-s + 1.32·4-s − 0.630·5-s + 7-s − 1.23·8-s − 1.14·10-s + 2.60·11-s − 4.34·13-s + 1.82·14-s − 4.89·16-s + 1.67·17-s + 7.62·19-s − 0.834·20-s + 4.75·22-s − 4.50·23-s − 4.60·25-s − 7.92·26-s + 1.32·28-s + 1.00·29-s − 6.18·31-s − 6.45·32-s + 3.05·34-s − 0.630·35-s − 0.667·37-s + 13.9·38-s + 0.778·40-s − 8.44·41-s + ⋯ |
L(s) = 1 | + 1.28·2-s + 0.661·4-s − 0.282·5-s + 0.377·7-s − 0.436·8-s − 0.363·10-s + 0.786·11-s − 1.20·13-s + 0.487·14-s − 1.22·16-s + 0.406·17-s + 1.74·19-s − 0.186·20-s + 1.01·22-s − 0.939·23-s − 0.920·25-s − 1.55·26-s + 0.250·28-s + 0.186·29-s − 1.11·31-s − 1.14·32-s + 0.523·34-s − 0.106·35-s − 0.109·37-s + 2.25·38-s + 0.123·40-s − 1.31·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 - 1.82T + 2T^{2} \) |
| 5 | \( 1 + 0.630T + 5T^{2} \) |
| 11 | \( 1 - 2.60T + 11T^{2} \) |
| 13 | \( 1 + 4.34T + 13T^{2} \) |
| 17 | \( 1 - 1.67T + 17T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 + 4.50T + 23T^{2} \) |
| 29 | \( 1 - 1.00T + 29T^{2} \) |
| 31 | \( 1 + 6.18T + 31T^{2} \) |
| 37 | \( 1 + 0.667T + 37T^{2} \) |
| 41 | \( 1 + 8.44T + 41T^{2} \) |
| 43 | \( 1 - 0.767T + 43T^{2} \) |
| 47 | \( 1 + 1.27T + 47T^{2} \) |
| 53 | \( 1 + 9.23T + 53T^{2} \) |
| 59 | \( 1 + 3.99T + 59T^{2} \) |
| 61 | \( 1 + 6.45T + 61T^{2} \) |
| 67 | \( 1 - 5.31T + 67T^{2} \) |
| 71 | \( 1 - 9.52T + 71T^{2} \) |
| 73 | \( 1 - 4.29T + 73T^{2} \) |
| 79 | \( 1 + 2.53T + 79T^{2} \) |
| 83 | \( 1 - 4.96T + 83T^{2} \) |
| 89 | \( 1 - 0.366T + 89T^{2} \) |
| 97 | \( 1 + 10.1T + 97T^{2} \) |
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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
−7.45592824963754769267812834542, −6.63690831134507308911297379105, −5.91583626562855160802795882664, −5.16289310999395099612536837109, −4.81875917247986681868483345526, −3.81042581503503199371985553815, −3.46698042487706113217306921652, −2.47929709767864894918569567434, −1.51399440962788534261189351418, 0,
1.51399440962788534261189351418, 2.47929709767864894918569567434, 3.46698042487706113217306921652, 3.81042581503503199371985553815, 4.81875917247986681868483345526, 5.16289310999395099612536837109, 5.91583626562855160802795882664, 6.63690831134507308911297379105, 7.45592824963754769267812834542