| L(s) = 1 | − 16.5·2-s + 147.·4-s − 174.·5-s + 1.11e3·7-s − 319.·8-s + 2.90e3·10-s − 3.45e3·11-s − 368.·13-s − 1.84e4·14-s − 1.35e4·16-s − 1.20e4·17-s − 2.03e4·19-s − 2.57e4·20-s + 5.73e4·22-s − 2.76e4·23-s − 4.75e4·25-s + 6.11e3·26-s + 1.63e5·28-s − 1.41e5·29-s + 2.24e5·31-s + 2.65e5·32-s + 1.99e5·34-s − 1.94e5·35-s + 4.33e5·37-s + 3.38e5·38-s + 5.58e4·40-s − 2.86e5·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.15·4-s − 0.626·5-s + 1.22·7-s − 0.220·8-s + 0.918·10-s − 0.783·11-s − 0.0465·13-s − 1.79·14-s − 0.827·16-s − 0.593·17-s − 0.682·19-s − 0.720·20-s + 1.14·22-s − 0.474·23-s − 0.608·25-s + 0.0682·26-s + 1.40·28-s − 1.08·29-s + 1.35·31-s + 1.43·32-s + 0.870·34-s − 0.766·35-s + 1.40·37-s + 1.00·38-s + 0.138·40-s − 0.648·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.5299311963\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5299311963\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 16.5T + 128T^{2} \) |
| 5 | \( 1 + 174.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.11e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 3.45e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 368.T + 6.27e7T^{2} \) |
| 17 | \( 1 + 1.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.03e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.76e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.41e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.24e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 4.33e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 2.86e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.52e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 4.49e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.41e5T + 1.17e12T^{2} \) |
| 61 | \( 1 + 5.22e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.89e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.16e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.37e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.64e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 6.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 5.43e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.91e6T + 8.07e13T^{2} \) |
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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
−9.666601090830459809544660863360, −8.608882543825735759654107322752, −8.014148963520106534912618274877, −7.57879508003070985703247130640, −6.39115500435111461955443927444, −4.98397128970776515712810910173, −4.12781818744461489899945597662, −2.44584554753064480506855746590, −1.58920051974759745657278169377, −0.40140644266884760814970994594,
0.40140644266884760814970994594, 1.58920051974759745657278169377, 2.44584554753064480506855746590, 4.12781818744461489899945597662, 4.98397128970776515712810910173, 6.39115500435111461955443927444, 7.57879508003070985703247130640, 8.014148963520106534912618274877, 8.608882543825735759654107322752, 9.666601090830459809544660863360
