| L(s) = 1 | + 269.·2-s + 3.97e4·4-s + 2.96e4·5-s − 1.74e6·7-s + 1.87e6·8-s + 7.97e6·10-s − 1.00e8·11-s − 9.38e7·13-s − 4.69e8·14-s − 7.96e8·16-s + 1.06e9·17-s − 4.26e8·19-s + 1.17e9·20-s − 2.70e10·22-s + 1.12e10·23-s − 2.96e10·25-s − 2.52e10·26-s − 6.92e10·28-s − 2.67e10·29-s − 2.36e11·31-s − 2.76e11·32-s + 2.85e11·34-s − 5.16e10·35-s + 2.15e11·37-s − 1.14e11·38-s + 5.55e10·40-s + 7.40e11·41-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 1.21·4-s + 0.169·5-s − 0.800·7-s + 0.316·8-s + 0.252·10-s − 1.55·11-s − 0.414·13-s − 1.19·14-s − 0.741·16-s + 0.627·17-s − 0.109·19-s + 0.205·20-s − 2.31·22-s + 0.685·23-s − 0.971·25-s − 0.617·26-s − 0.970·28-s − 0.287·29-s − 1.54·31-s − 1.42·32-s + 0.933·34-s − 0.135·35-s + 0.372·37-s − 0.162·38-s + 0.0536·40-s + 0.593·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 - 269.T + 3.27e4T^{2} \) |
| 5 | \( 1 - 2.96e4T + 3.05e10T^{2} \) |
| 7 | \( 1 + 1.74e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 1.00e8T + 4.17e15T^{2} \) |
| 13 | \( 1 + 9.38e7T + 5.11e16T^{2} \) |
| 17 | \( 1 - 1.06e9T + 2.86e18T^{2} \) |
| 19 | \( 1 + 4.26e8T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.12e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 2.67e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 2.36e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 2.15e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 7.40e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 3.44e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 5.09e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 9.29e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 2.26e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.91e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 8.17e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.01e14T + 5.87e27T^{2} \) |
| 73 | \( 1 - 8.97e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.90e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.73e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 4.82e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 2.09e13T + 6.33e29T^{2} \) |
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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
−13.10070363185626613391695336924, −12.59328627843039872794403701358, −11.02568211431018052337644394610, −9.561912670945579669672539821276, −7.53106613004233812535391790807, −6.01032813804225600328598345430, −5.02886622871136597184737912067, −3.49019362459814829087653600775, −2.40212748807782095522340241635, 0,
2.40212748807782095522340241635, 3.49019362459814829087653600775, 5.02886622871136597184737912067, 6.01032813804225600328598345430, 7.53106613004233812535391790807, 9.561912670945579669672539821276, 11.02568211431018052337644394610, 12.59328627843039872794403701358, 13.10070363185626613391695336924
