| L(s) = 1 | − 1.72e3·2-s + 8.88e5·4-s + 4.15e7·5-s + 5.38e8·7-s + 2.08e9·8-s − 7.17e10·10-s + 6.41e10·11-s − 1.30e11·13-s − 9.30e11·14-s − 5.47e12·16-s − 8.24e12·17-s + 1.34e13·19-s + 3.68e13·20-s − 1.10e14·22-s + 2.33e14·23-s + 1.24e15·25-s + 2.26e14·26-s + 4.78e14·28-s + 2.02e15·29-s − 6.86e15·31-s + 5.07e15·32-s + 1.42e16·34-s + 2.23e16·35-s + 3.44e15·37-s − 2.33e16·38-s + 8.66e16·40-s + 2.18e16·41-s + ⋯ |
| L(s) = 1 | − 1.19·2-s + 0.423·4-s + 1.90·5-s + 0.720·7-s + 0.687·8-s − 2.26·10-s + 0.745·11-s − 0.263·13-s − 0.859·14-s − 1.24·16-s − 0.991·17-s + 0.504·19-s + 0.805·20-s − 0.889·22-s + 1.17·23-s + 2.61·25-s + 0.314·26-s + 0.305·28-s + 0.893·29-s − 1.50·31-s + 0.797·32-s + 1.18·34-s + 1.36·35-s + 0.117·37-s − 0.602·38-s + 1.30·40-s + 0.254·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.632099049\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.632099049\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 + 27 p^{6} T + p^{21} T^{2} \) |
| 5 | \( 1 - 8302554 p T + p^{21} T^{2} \) |
| 7 | \( 1 - 76918544 p T + p^{21} T^{2} \) |
| 11 | \( 1 - 64113040188 T + p^{21} T^{2} \) |
| 13 | \( 1 + 10075392922 p T + p^{21} T^{2} \) |
| 17 | \( 1 + 8242029723618 T + p^{21} T^{2} \) |
| 19 | \( 1 - 710110618580 p T + p^{21} T^{2} \) |
| 23 | \( 1 - 233184825844776 T + p^{21} T^{2} \) |
| 29 | \( 1 - 2024562031123770 T + p^{21} T^{2} \) |
| 31 | \( 1 + 6869194988701768 T + p^{21} T^{2} \) |
| 37 | \( 1 - 3443998107027638 T + p^{21} T^{2} \) |
| 41 | \( 1 - 21842403084625158 T + p^{21} T^{2} \) |
| 43 | \( 1 + 71792816814133756 T + p^{21} T^{2} \) |
| 47 | \( 1 + 283544719418655648 T + p^{21} T^{2} \) |
| 53 | \( 1 - 2172285419049898146 T + p^{21} T^{2} \) |
| 59 | \( 1 + 1534831476719068260 T + p^{21} T^{2} \) |
| 61 | \( 1 - 4311589520797626062 T + p^{21} T^{2} \) |
| 67 | \( 1 - 9243910904037307868 T + p^{21} T^{2} \) |
| 71 | \( 1 - 20387361256404760728 T + p^{21} T^{2} \) |
| 73 | \( 1 - 16617754439328636074 T + p^{21} T^{2} \) |
| 79 | \( 1 - 67940304745507627880 T + p^{21} T^{2} \) |
| 83 | \( 1 + 39503732340682314684 T + p^{21} T^{2} \) |
| 89 | \( 1 + 41611676186839694490 T + p^{21} T^{2} \) |
| 97 | \( 1 - 57181473208903260098 T + p^{21} T^{2} \) |
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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
−16.75420856182694251736555482171, −14.44135274482438417288439128494, −13.28909052405761770811830471507, −10.95151861394262444236288511775, −9.666556858723593195503851030597, −8.781920797525711213730155976802, −6.81529621360294006877832505656, −5.04768469476127118089759234888, −2.10879241321839161045001494973, −1.10782541805633434444756377034,
1.10782541805633434444756377034, 2.10879241321839161045001494973, 5.04768469476127118089759234888, 6.81529621360294006877832505656, 8.781920797525711213730155976802, 9.666556858723593195503851030597, 10.95151861394262444236288511775, 13.28909052405761770811830471507, 14.44135274482438417288439128494, 16.75420856182694251736555482171
