| L(s) = 1 | + 19.4·2-s + 250·4-s + 388.·5-s − 1.26e3·7-s + 2.37e3·8-s + 7.56e3·10-s + 1.47e3·11-s + 9.58e3·13-s − 2.45e4·14-s + 1.41e4·16-s − 2.12e4·17-s − 2.19e4·19-s + 9.72e4·20-s + 2.87e4·22-s − 8.59e4·23-s + 7.30e4·25-s + 1.86e5·26-s − 3.15e5·28-s − 3.23e4·29-s − 5.09e4·31-s − 2.91e4·32-s − 4.12e5·34-s − 4.90e5·35-s + 2.46e5·37-s − 4.26e5·38-s + 9.22e5·40-s + 6.10e5·41-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 1.95·4-s + 1.39·5-s − 1.38·7-s + 1.63·8-s + 2.39·10-s + 0.334·11-s + 1.20·13-s − 2.38·14-s + 0.861·16-s − 1.04·17-s − 0.733·19-s + 2.71·20-s + 0.575·22-s − 1.47·23-s + 0.935·25-s + 2.07·26-s − 2.71·28-s − 0.246·29-s − 0.306·31-s − 0.157·32-s − 1.80·34-s − 1.93·35-s + 0.799·37-s − 1.26·38-s + 2.27·40-s + 1.38·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(4.547123798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.547123798\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 19.4T + 128T^{2} \) |
| 5 | \( 1 - 388.T + 7.81e4T^{2} \) |
| 7 | \( 1 + 1.26e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 1.47e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 9.58e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.12e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.19e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.59e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 3.23e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 5.09e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 2.46e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.10e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.15e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.25e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.27e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 9.64e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 4.97e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 1.33e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 9.01e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.25e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.07e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.30e7T + 4.42e13T^{2} \) |
| 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
−15.54151922671982725401025626497, −14.06635388479048649217379819389, −13.35423197678409463548671907688, −12.58329156384406705025285111853, −10.86278126331819736099924331314, −9.358301319847822476760555356046, −6.42555123176556710940437182330, −5.95823610460272958072634246998, −3.93866050416968271871970749261, −2.29744490774972589050450996139,
2.29744490774972589050450996139, 3.93866050416968271871970749261, 5.95823610460272958072634246998, 6.42555123176556710940437182330, 9.358301319847822476760555356046, 10.86278126331819736099924331314, 12.58329156384406705025285111853, 13.35423197678409463548671907688, 14.06635388479048649217379819389, 15.54151922671982725401025626497
