| L(s) = 1 | + 19.9·2-s + 56.3·3-s + 270.·4-s + 1.53·5-s + 1.12e3·6-s + 343·7-s + 2.84e3·8-s + 984.·9-s + 30.5·10-s + 702.·11-s + 1.52e4·12-s + 2.19e3·13-s + 6.84e3·14-s + 86.1·15-s + 2.21e4·16-s − 1.36e4·17-s + 1.96e4·18-s + 9.52e3·19-s + 413.·20-s + 1.93e4·21-s + 1.40e4·22-s + 6.87e4·23-s + 1.60e5·24-s − 7.81e4·25-s + 4.38e4·26-s − 6.77e4·27-s + 9.27e4·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s + 1.20·3-s + 2.11·4-s + 0.00547·5-s + 2.12·6-s + 0.377·7-s + 1.96·8-s + 0.450·9-s + 0.00966·10-s + 0.159·11-s + 2.54·12-s + 0.277·13-s + 0.666·14-s + 0.00659·15-s + 1.35·16-s − 0.673·17-s + 0.794·18-s + 0.318·19-s + 0.0115·20-s + 0.455·21-s + 0.280·22-s + 1.17·23-s + 2.36·24-s − 0.999·25-s + 0.489·26-s − 0.662·27-s + 0.798·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(8.429901019\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.429901019\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 - 2.19e3T \) |
| good | 2 | \( 1 - 19.9T + 128T^{2} \) |
| 3 | \( 1 - 56.3T + 2.18e3T^{2} \) |
| 5 | \( 1 - 1.53T + 7.81e4T^{2} \) |
| 11 | \( 1 - 702.T + 1.94e7T^{2} \) |
| 17 | \( 1 + 1.36e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.52e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.87e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 8.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 7.84e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 1.72e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.18e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 2.38e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 8.91e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 5.55e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.18e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.75e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.17e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.77e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 7.57e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 3.59e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.17e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.92e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.38e6T + 8.07e13T^{2} \) |
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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
−13.19174521898087199689741214694, −11.89437302487455603860484725828, −10.96331972677873169831347136723, −9.260706243731600531667536884510, −7.968996386376129932003694027255, −6.72945673559640612410704415883, −5.33642194396455064757164785528, −4.06316704120972146876320170989, −3.05642196884090982834010073390, −1.90825323555893724232474652023,
1.90825323555893724232474652023, 3.05642196884090982834010073390, 4.06316704120972146876320170989, 5.33642194396455064757164785528, 6.72945673559640612410704415883, 7.968996386376129932003694027255, 9.260706243731600531667536884510, 10.96331972677873169831347136723, 11.89437302487455603860484725828, 13.19174521898087199689741214694
