| L(s) = 1 | − 5.91·2-s + 2.98·4-s + 22.5·5-s + 90.8·7-s + 171.·8-s − 133.·10-s − 272.·11-s − 279.·13-s − 537.·14-s − 1.11e3·16-s − 1.47e3·17-s − 242.·19-s + 67.1·20-s + 1.61e3·22-s + 132.·23-s − 2.61e3·25-s + 1.65e3·26-s + 270.·28-s + 8.27e3·29-s + 7.61e3·31-s + 1.07e3·32-s + 8.71e3·34-s + 2.04e3·35-s + 1.14e4·37-s + 1.43e3·38-s + 3.86e3·40-s − 5.79e3·41-s + ⋯ |
| L(s) = 1 | − 1.04·2-s + 0.0931·4-s + 0.402·5-s + 0.700·7-s + 0.948·8-s − 0.420·10-s − 0.678·11-s − 0.459·13-s − 0.732·14-s − 1.08·16-s − 1.23·17-s − 0.153·19-s + 0.0375·20-s + 0.709·22-s + 0.0523·23-s − 0.837·25-s + 0.480·26-s + 0.0652·28-s + 1.82·29-s + 1.42·31-s + 0.185·32-s + 1.29·34-s + 0.281·35-s + 1.37·37-s + 0.160·38-s + 0.381·40-s − 0.538·41-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
| \(L(3)\) |
\(\approx\) |
\(0.9726365958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9726365958\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.91T + 32T^{2} \) |
| 5 | \( 1 - 22.5T + 3.12e3T^{2} \) |
| 7 | \( 1 - 90.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 272.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 279.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.47e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 242.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 132.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.61e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.14e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.79e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.28e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.98e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.68e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 6.20e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.01e5T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.58e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.39e5T + 8.58e9T^{2} \) |
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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
−9.863868186395374787722683882431, −8.496983097725821039558409234521, −8.328125559059727253815521525557, −7.25365137440528228122603121518, −6.30756142061977164507904702374, −4.96255119646243634566764708963, −4.42417443246279224767750027536, −2.64615964026355700746003972826, −1.71274744299944881078118295849, −0.54578428529859423796247105060,
0.54578428529859423796247105060, 1.71274744299944881078118295849, 2.64615964026355700746003972826, 4.42417443246279224767750027536, 4.96255119646243634566764708963, 6.30756142061977164507904702374, 7.25365137440528228122603121518, 8.328125559059727253815521525557, 8.496983097725821039558409234521, 9.863868186395374787722683882431
