| L(s) = 1 | − 1.44·2-s − 29.9·4-s − 31.6·5-s − 172.·7-s + 89.2·8-s + 45.6·10-s + 1.42·11-s + 286.·13-s + 248.·14-s + 828.·16-s − 190.·17-s − 93.7·19-s + 946.·20-s − 2.04·22-s − 3.59e3·23-s − 2.12e3·25-s − 413.·26-s + 5.15e3·28-s − 6.72e3·29-s − 2.79e3·31-s − 4.05e3·32-s + 275.·34-s + 5.45e3·35-s − 5.98e3·37-s + 135.·38-s − 2.82e3·40-s − 1.75e4·41-s + ⋯ |
| L(s) = 1 | − 0.254·2-s − 0.935·4-s − 0.566·5-s − 1.32·7-s + 0.493·8-s + 0.144·10-s + 0.00354·11-s + 0.470·13-s + 0.338·14-s + 0.809·16-s − 0.160·17-s − 0.0595·19-s + 0.529·20-s − 0.000902·22-s − 1.41·23-s − 0.679·25-s − 0.119·26-s + 1.24·28-s − 1.48·29-s − 0.522·31-s − 0.699·32-s + 0.0408·34-s + 0.752·35-s − 0.718·37-s + 0.0151·38-s − 0.279·40-s − 1.62·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.05432890454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05432890454\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 1.44T + 32T^{2} \) |
| 5 | \( 1 + 31.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 172.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 1.42T + 1.61e5T^{2} \) |
| 13 | \( 1 - 286.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 190.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 93.7T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.59e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.72e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.79e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.98e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.75e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.91e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.11e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.66e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.84e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.80e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.39e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.32e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.08e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.01e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.43e4T + 8.58e9T^{2} \) |
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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
−9.801841080230044898458267667778, −8.741520637579590066531019133857, −8.110083109353152087768615482388, −7.08172769778061783681892487975, −6.11501702975761006559862911898, −5.13204987788764513463651246604, −3.81506510188613022084812134526, −3.50491670163381619229327363305, −1.75193812807670926207857549299, −0.11103868692428728095878591743,
0.11103868692428728095878591743, 1.75193812807670926207857549299, 3.50491670163381619229327363305, 3.81506510188613022084812134526, 5.13204987788764513463651246604, 6.11501702975761006559862911898, 7.08172769778061783681892487975, 8.110083109353152087768615482388, 8.741520637579590066531019133857, 9.801841080230044898458267667778
