| L(s) = 1 | − 225.·2-s + 1.78e4·4-s − 3.38e5·5-s + 6.26e5·7-s + 3.35e6·8-s + 7.60e7·10-s − 5.98e7·11-s + 1.31e8·13-s − 1.40e8·14-s − 1.34e9·16-s + 9.05e8·17-s + 3.66e9·19-s − 6.03e9·20-s + 1.34e10·22-s + 2.86e10·23-s + 8.38e10·25-s − 2.94e10·26-s + 1.11e10·28-s − 1.63e11·29-s − 1.17e10·31-s + 1.91e11·32-s − 2.03e11·34-s − 2.11e11·35-s + 6.27e11·37-s − 8.24e11·38-s − 1.13e12·40-s − 5.13e11·41-s + ⋯ |
| L(s) = 1 | − 1.24·2-s + 0.545·4-s − 1.93·5-s + 0.287·7-s + 0.565·8-s + 2.40·10-s − 0.926·11-s + 0.579·13-s − 0.357·14-s − 1.24·16-s + 0.535·17-s + 0.940·19-s − 1.05·20-s + 1.15·22-s + 1.75·23-s + 2.74·25-s − 0.720·26-s + 0.156·28-s − 1.75·29-s − 0.0764·31-s + 0.985·32-s − 0.665·34-s − 0.556·35-s + 1.08·37-s − 1.16·38-s − 1.09·40-s − 0.412·41-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 | \( 1 + 225.T + 3.27e4T^{2} \) |
| 5 | \( 1 + 3.38e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 6.26e5T + 4.74e12T^{2} \) |
| 11 | \( 1 + 5.98e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.31e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 9.05e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 3.66e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.86e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.63e11T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.17e10T + 2.34e22T^{2} \) |
| 37 | \( 1 - 6.27e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 5.13e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 3.28e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 1.43e12T + 1.20e25T^{2} \) |
| 53 | \( 1 - 2.56e12T + 7.31e25T^{2} \) |
| 59 | \( 1 + 2.85e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.68e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 4.75e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.04e14T + 5.87e27T^{2} \) |
| 73 | \( 1 + 8.48e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 2.07e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.60e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 3.31e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.15e15T + 6.33e29T^{2} \) |
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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
−13.05058292495260462524273716040, −11.49494039200145908720027309218, −10.77966398670126750310820409873, −9.096079873228300557454317209759, −7.919124078567173906617201703797, −7.39207651445010281029194968688, −4.81983187781338394732395635050, −3.31047861056560793721377912457, −1.06949547497464029899679170480, 0,
1.06949547497464029899679170480, 3.31047861056560793721377912457, 4.81983187781338394732395635050, 7.39207651445010281029194968688, 7.919124078567173906617201703797, 9.096079873228300557454317209759, 10.77966398670126750310820409873, 11.49494039200145908720027309218, 13.05058292495260462524273716040
