| L(s) = 1 | + 2.08·2-s + 3-s + 2.35·4-s − 3.16·5-s + 2.08·6-s + 0.308·7-s + 0.749·8-s + 9-s − 6.61·10-s − 5.34·11-s + 2.35·12-s + 5.25·13-s + 0.643·14-s − 3.16·15-s − 3.15·16-s + 17-s + 2.08·18-s + 2.21·19-s − 7.47·20-s + 0.308·21-s − 11.1·22-s − 1.72·23-s + 0.749·24-s + 5.04·25-s + 10.9·26-s + 27-s + 0.727·28-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 0.577·3-s + 1.17·4-s − 1.41·5-s + 0.852·6-s + 0.116·7-s + 0.265·8-s + 0.333·9-s − 2.09·10-s − 1.61·11-s + 0.681·12-s + 1.45·13-s + 0.172·14-s − 0.818·15-s − 0.788·16-s + 0.242·17-s + 0.492·18-s + 0.509·19-s − 1.67·20-s + 0.0672·21-s − 2.38·22-s − 0.360·23-s + 0.153·24-s + 1.00·25-s + 2.15·26-s + 0.192·27-s + 0.137·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 7 | \( 1 - 0.308T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 - 5.25T + 13T^{2} \) |
| 19 | \( 1 - 2.21T + 19T^{2} \) |
| 23 | \( 1 + 1.72T + 23T^{2} \) |
| 29 | \( 1 + 7.33T + 29T^{2} \) |
| 31 | \( 1 + 3.47T + 31T^{2} \) |
| 37 | \( 1 + 3.75T + 37T^{2} \) |
| 41 | \( 1 + 10.5T + 41T^{2} \) |
| 43 | \( 1 + 0.578T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + 9.54T + 53T^{2} \) |
| 59 | \( 1 + 8.21T + 59T^{2} \) |
| 61 | \( 1 + 12.8T + 61T^{2} \) |
| 67 | \( 1 + 0.0935T + 67T^{2} \) |
| 71 | \( 1 - 2.20T + 71T^{2} \) |
| 73 | \( 1 - 7.90T + 73T^{2} \) |
| 83 | \( 1 - 10.1T + 83T^{2} \) |
| 89 | \( 1 + 9.77T + 89T^{2} \) |
| 97 | \( 1 - 8.88T + 97T^{2} \) |
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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
−7.84321399569796308753823467392, −7.48422368404264464326396323918, −6.48327120630855304315138454738, −5.58201055623203396611223026679, −4.96436963447547798508462035818, −4.11572720158113427785003026269, −3.45046554304166511190641334741, −3.08913403341527315256076508931, −1.82611447778146009719357010658, 0,
1.82611447778146009719357010658, 3.08913403341527315256076508931, 3.45046554304166511190641334741, 4.11572720158113427785003026269, 4.96436963447547798508462035818, 5.58201055623203396611223026679, 6.48327120630855304315138454738, 7.48422368404264464326396323918, 7.84321399569796308753823467392
