| L(s) = 1 | − 10.5i·2-s + 145.·4-s − 368. i·5-s − 2.40e3·7-s − 4.22e3i·8-s − 3.88e3·10-s + 1.83e4i·11-s + 3.11e4·13-s + 2.53e4i·14-s − 7.29e3·16-s − 8.97e4i·17-s − 9.08e3·19-s − 5.35e4i·20-s + 1.93e5·22-s − 9.44e4i·23-s + ⋯ |
| L(s) = 1 | − 0.657i·2-s + 0.567·4-s − 0.590i·5-s − 1.00·7-s − 1.03i·8-s − 0.388·10-s + 1.25i·11-s + 1.08·13-s + 0.659i·14-s − 0.111·16-s − 1.07i·17-s − 0.0697·19-s − 0.334i·20-s + 0.825·22-s − 0.337i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.572405132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.572405132\) |
| \(L(5)\) |
|
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 | \( 1 + 10.5iT - 256T^{2} \) |
| 5 | \( 1 + 368. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 2.40e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.83e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.11e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 8.97e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 9.08e3T + 1.69e10T^{2} \) |
| 23 | \( 1 + 9.44e4iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 8.25e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 4.04e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 8.45e5T + 3.51e12T^{2} \) |
| 41 | \( 1 - 3.91e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 5.56e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 4.77e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 3.73e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.32e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 4.08e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 1.85e7T + 4.06e14T^{2} \) |
| 71 | \( 1 - 3.35e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 3.83e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.14e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.82e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.22e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.89e7T + 7.83e15T^{2} \) |
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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
−10.03637240815249974591619969496, −9.626189744061862856141923670554, −8.355172058157462777006457208019, −7.00387150073617821127139055588, −6.36012031471540902534853377147, −4.87288458716997012484676429534, −3.66118675993122920569596018132, −2.61585612269333969325968049994, −1.45115808153278826709562843388, −0.32720393087514106622757087084,
1.34325234671677864256125759484, 2.90036962054611391535693339631, 3.61379227949146065808411504874, 5.55780348164306471652169393701, 6.31185047185332718579659889712, 6.90713400903771116640969829368, 8.189597376121895999407414330832, 8.939946014247635082547223691279, 10.52318810856315991713373471265, 10.88124633093033242666784650285
