| L(s) = 1 | + 10.6·2-s + 82.1·4-s − 25.8·5-s − 115.·7-s + 535.·8-s − 276.·10-s + 63.0·11-s − 797.·13-s − 1.23e3·14-s + 3.09e3·16-s + 1.14e3·17-s − 752.·19-s − 2.12e3·20-s + 674.·22-s − 265.·23-s − 2.45e3·25-s − 8.51e3·26-s − 9.47e3·28-s + 7.18e3·29-s + 2.69e3·31-s + 1.59e4·32-s + 1.22e4·34-s + 2.98e3·35-s + 1.11e4·37-s − 8.03e3·38-s − 1.38e4·40-s − 9.94e3·41-s + ⋯ |
| L(s) = 1 | + 1.88·2-s + 2.56·4-s − 0.462·5-s − 0.889·7-s + 2.96·8-s − 0.873·10-s + 0.157·11-s − 1.30·13-s − 1.68·14-s + 3.02·16-s + 0.962·17-s − 0.477·19-s − 1.18·20-s + 0.296·22-s − 0.104·23-s − 0.786·25-s − 2.47·26-s − 2.28·28-s + 1.58·29-s + 0.503·31-s + 2.75·32-s + 1.81·34-s + 0.411·35-s + 1.34·37-s − 0.902·38-s − 1.36·40-s − 0.924·41-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
| \(L(3)\) |
\(\approx\) |
\(3.597140559\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.597140559\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 10.6T + 32T^{2} \) |
| 5 | \( 1 + 25.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 115.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 63.0T + 1.61e5T^{2} \) |
| 13 | \( 1 + 797.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 752.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 265.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.18e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.11e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.94e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.96e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.02e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.57e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.02e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.07e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.42e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.11e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 7.57e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.65e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.17e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 4.53e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 376.T + 8.58e9T^{2} \) |
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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
−15.84763133463860043477732310787, −14.87716481169449254114110107138, −13.79865952975042329270695140637, −12.52353871001482257668814694116, −11.86800447323758941027856328906, −10.17461015324595017177747965504, −7.43928491821176510645796548479, −6.06664170951239721015027573340, −4.42910121150512498809314094435, −2.88967846694948628026951494279,
2.88967846694948628026951494279, 4.42910121150512498809314094435, 6.06664170951239721015027573340, 7.43928491821176510645796548479, 10.17461015324595017177747965504, 11.86800447323758941027856328906, 12.52353871001482257668814694116, 13.79865952975042329270695140637, 14.87716481169449254114110107138, 15.84763133463860043477732310787
