| L(s) = 1 | − 32.3·2-s + 533.·4-s + 2.07e3·5-s − 1.05e4·7-s − 692.·8-s − 6.69e4·10-s − 3.16e4·11-s + 4.76e4·13-s + 3.40e5·14-s − 2.50e5·16-s + 3.44e4·17-s + 7.63e5·19-s + 1.10e6·20-s + 1.02e6·22-s + 1.48e6·23-s + 2.33e6·25-s − 1.53e6·26-s − 5.61e6·28-s + 4.80e6·29-s − 7.82e6·31-s + 8.46e6·32-s − 1.11e6·34-s − 2.17e7·35-s + 9.43e6·37-s − 2.46e7·38-s − 1.43e6·40-s + 1.22e7·41-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.04·4-s + 1.48·5-s − 1.65·7-s − 0.0598·8-s − 2.11·10-s − 0.651·11-s + 0.462·13-s + 2.36·14-s − 0.956·16-s + 0.100·17-s + 1.34·19-s + 1.54·20-s + 0.930·22-s + 1.10·23-s + 1.19·25-s − 0.660·26-s − 1.72·28-s + 1.26·29-s − 1.52·31-s + 1.42·32-s − 0.142·34-s − 2.45·35-s + 0.827·37-s − 1.91·38-s − 0.0886·40-s + 0.679·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.9137539946\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9137539946\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 32.3T + 512T^{2} \) |
| 5 | \( 1 - 2.07e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.16e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.76e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.44e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.63e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.48e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.80e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.43e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.22e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.74e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 1.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 5.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 2.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.45e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.81e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.53e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.55e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 5.01e7T + 1.86e17T^{2} \) |
| 89 | \( 1 + 1.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.49e8T + 7.60e17T^{2} \) |
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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
−15.89901856449663434134751400170, −13.78268760694138106283696087720, −12.87906360041761217987777188142, −10.68189444762399446408471620669, −9.724076735022781039898880468487, −9.084182242151970970500206929957, −7.15873236176367245838248939618, −5.80849503420243861638131854253, −2.69844860864517553961576996314, −0.900555849938266409553346313049,
0.900555849938266409553346313049, 2.69844860864517553961576996314, 5.80849503420243861638131854253, 7.15873236176367245838248939618, 9.084182242151970970500206929957, 9.724076735022781039898880468487, 10.68189444762399446408471620669, 12.87906360041761217987777188142, 13.78268760694138106283696087720, 15.89901856449663434134751400170
