| L(s) = 1 | + 21.7·2-s + 346.·4-s + 409.·5-s + 476.·7-s + 4.76e3·8-s + 8.93e3·10-s − 6.86e3·11-s − 8.27e3·13-s + 1.03e4·14-s + 5.94e4·16-s + 1.61e4·17-s + 2.75e4·19-s + 1.42e5·20-s − 1.49e5·22-s + 6.91e4·23-s + 8.99e4·25-s − 1.80e5·26-s + 1.65e5·28-s + 1.11e5·29-s − 1.69e5·31-s + 6.84e5·32-s + 3.51e5·34-s + 1.95e5·35-s − 6.01e4·37-s + 5.99e5·38-s + 1.95e6·40-s + 8.13e5·41-s + ⋯ |
| L(s) = 1 | + 1.92·2-s + 2.70·4-s + 1.46·5-s + 0.525·7-s + 3.28·8-s + 2.82·10-s − 1.55·11-s − 1.04·13-s + 1.01·14-s + 3.62·16-s + 0.795·17-s + 0.920·19-s + 3.97·20-s − 2.99·22-s + 1.18·23-s + 1.15·25-s − 2.01·26-s + 1.42·28-s + 0.848·29-s − 1.02·31-s + 3.69·32-s + 1.53·34-s + 0.770·35-s − 0.195·37-s + 1.77·38-s + 4.82·40-s + 1.84·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 387 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(11.86758982\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.86758982\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 43 | \( 1 - 7.95e4T \) |
| good | 2 | \( 1 - 21.7T + 128T^{2} \) |
| 5 | \( 1 - 409.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 476.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 6.86e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 8.27e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.61e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 2.75e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 6.91e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.11e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.69e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 6.01e4T + 9.49e10T^{2} \) |
| 41 | \( 1 - 8.13e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 6.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.65e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.43e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.53e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.27e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 8.21e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.08e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 6.74e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.81e5T + 2.71e13T^{2} \) |
| 89 | \( 1 + 9.40e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 6.78e6T + 8.07e13T^{2} \) |
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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
−10.47374823103088234520419213418, −9.578989126243302171812670409074, −7.77820288808684370344455216992, −7.06108391257291230484937709315, −5.75129161595720956948528642621, −5.35634946333702314801875059227, −4.64889012149631846681989404849, −2.97911984346769078873775151760, −2.46449710721727794319796426685, −1.36151408313757456544216140126,
1.36151408313757456544216140126, 2.46449710721727794319796426685, 2.97911984346769078873775151760, 4.64889012149631846681989404849, 5.35634946333702314801875059227, 5.75129161595720956948528642621, 7.06108391257291230484937709315, 7.77820288808684370344455216992, 9.578989126243302171812670409074, 10.47374823103088234520419213418
