| L(s) = 1 | + 1.68·2-s − 29.1·4-s − 97.8·5-s + 107.·7-s − 103.·8-s − 164.·10-s − 458.·11-s + 93.2·13-s + 180.·14-s + 759.·16-s + 931.·17-s − 2.08e3·19-s + 2.85e3·20-s − 773.·22-s − 229.·23-s + 6.44e3·25-s + 157.·26-s − 3.12e3·28-s − 5.05e3·29-s − 3.76e3·31-s + 4.57e3·32-s + 1.56e3·34-s − 1.05e4·35-s − 2.17e3·37-s − 3.51e3·38-s + 1.00e4·40-s − 1.56e4·41-s + ⋯ |
| L(s) = 1 | + 0.297·2-s − 0.911·4-s − 1.75·5-s + 0.827·7-s − 0.569·8-s − 0.521·10-s − 1.14·11-s + 0.153·13-s + 0.246·14-s + 0.741·16-s + 0.781·17-s − 1.32·19-s + 1.59·20-s − 0.340·22-s − 0.0904·23-s + 2.06·25-s + 0.0455·26-s − 0.754·28-s − 1.11·29-s − 0.702·31-s + 0.790·32-s + 0.232·34-s − 1.44·35-s − 0.261·37-s − 0.395·38-s + 0.996·40-s − 1.45·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 1.68T + 32T^{2} \) |
| 5 | \( 1 + 97.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 107.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 458.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 93.2T + 3.71e5T^{2} \) |
| 17 | \( 1 - 931.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.08e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 229.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 2.17e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.56e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.18e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.70e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.92e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 1.20e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 7.31e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.30e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.05e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.12e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.69e5T + 8.58e9T^{2} \) |
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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
−15.39502855705285227673997739201, −14.64578887722905512107798798470, −13.06551024353514213937485183024, −11.95841274257853507711149653012, −10.66063145895024305716026926176, −8.555372447077073065801821521005, −7.70720377342555246055284682169, −5.06501297354475561411474000335, −3.76248226708603125570294469909, 0,
3.76248226708603125570294469909, 5.06501297354475561411474000335, 7.70720377342555246055284682169, 8.555372447077073065801821521005, 10.66063145895024305716026926176, 11.95841274257853507711149653012, 13.06551024353514213937485183024, 14.64578887722905512107798798470, 15.39502855705285227673997739201
