| L(s) = 1 | + 21.3·2-s + 326.·4-s + 188.·5-s + 639.·7-s + 4.21e3·8-s + 4.01e3·10-s + 1.62e3·11-s + 1.13e3·13-s + 1.36e4·14-s + 4.81e4·16-s − 2.87e4·17-s + 4.33e4·19-s + 6.13e4·20-s + 3.47e4·22-s + 1.21e4·23-s − 4.26e4·25-s + 2.42e4·26-s + 2.08e5·28-s − 6.25e4·29-s + 2.25e4·31-s + 4.86e5·32-s − 6.11e5·34-s + 1.20e5·35-s − 2.77e5·37-s + 9.24e5·38-s + 7.94e5·40-s + 1.30e5·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.54·4-s + 0.673·5-s + 0.704·7-s + 2.91·8-s + 1.26·10-s + 0.369·11-s + 0.143·13-s + 1.32·14-s + 2.94·16-s − 1.41·17-s + 1.45·19-s + 1.71·20-s + 0.695·22-s + 0.208·23-s − 0.546·25-s + 0.270·26-s + 1.79·28-s − 0.476·29-s + 0.135·31-s + 2.62·32-s − 2.66·34-s + 0.474·35-s − 0.900·37-s + 2.73·38-s + 1.96·40-s + 0.294·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.461773329\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.461773329\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 23 | \( 1 - 1.21e4T \) |
| good | 2 | \( 1 - 21.3T + 128T^{2} \) |
| 5 | \( 1 - 188.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 639.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.62e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.13e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.87e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.33e4T + 8.93e8T^{2} \) |
| 29 | \( 1 + 6.25e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.25e4T + 2.75e10T^{2} \) |
| 37 | \( 1 + 2.77e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.30e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.85e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 9.93e4T + 5.06e11T^{2} \) |
| 53 | \( 1 - 6.36e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.08e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 2.40e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.05e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 2.78e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.20e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 4.98e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.57e6T + 8.07e13T^{2} \) |
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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
−11.43437212682400693112572934192, −10.65883204260138234324652579357, −9.239948152920729141314040580780, −7.64706419555000002411618052921, −6.62545353176085558807598566835, −5.66607427140930627475541371168, −4.82924949959668494839130570464, −3.77935890873445910561527117163, −2.48055058566585435036081982805, −1.49519529412475298142001204255,
1.49519529412475298142001204255, 2.48055058566585435036081982805, 3.77935890873445910561527117163, 4.82924949959668494839130570464, 5.66607427140930627475541371168, 6.62545353176085558807598566835, 7.64706419555000002411618052921, 9.239948152920729141314040580780, 10.65883204260138234324652579357, 11.43437212682400693112572934192
