| L(s) = 1 | − 4·2-s − 15.5·3-s + 16·4-s − 109.·5-s + 62.1·6-s + 49·7-s − 64·8-s − 1.94·9-s + 438.·10-s + 650.·11-s − 248.·12-s + 169·13-s − 196·14-s + 1.70e3·15-s + 256·16-s + 1.34e3·17-s + 7.79·18-s − 2.66e3·19-s − 1.75e3·20-s − 760.·21-s − 2.60e3·22-s + 1.44e3·23-s + 993.·24-s + 8.88e3·25-s − 676·26-s + 3.80e3·27-s + 784·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.995·3-s + 0.5·4-s − 1.96·5-s + 0.704·6-s + 0.377·7-s − 0.353·8-s − 0.00801·9-s + 1.38·10-s + 1.61·11-s − 0.497·12-s + 0.277·13-s − 0.267·14-s + 1.95·15-s + 0.250·16-s + 1.13·17-s + 0.00566·18-s − 1.69·19-s − 0.980·20-s − 0.376·21-s − 1.14·22-s + 0.568·23-s + 0.352·24-s + 2.84·25-s − 0.196·26-s + 1.00·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 182 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 4T \) |
| 7 | \( 1 - 49T \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 + 15.5T + 243T^{2} \) |
| 5 | \( 1 + 109.T + 3.12e3T^{2} \) |
| 11 | \( 1 - 650.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.34e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.66e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.44e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 791.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 4.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 6.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 7.85e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.09e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 7.28e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.29e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.32e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.52e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.87e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.17e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.96e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.23e4T + 8.58e9T^{2} \) |
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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
−11.28992784056441275137221153542, −10.68307754355158807392074146148, −9.004009825644976831188782771300, −8.202666793745011033078395217439, −7.16412305078910652557667791640, −6.19321732383801264809656579534, −4.57906818973859686868759385469, −3.50793479878521672691882986135, −1.11055537185174085601103602485, 0,
1.11055537185174085601103602485, 3.50793479878521672691882986135, 4.57906818973859686868759385469, 6.19321732383801264809656579534, 7.16412305078910652557667791640, 8.202666793745011033078395217439, 9.004009825644976831188782771300, 10.68307754355158807392074146148, 11.28992784056441275137221153542
