| L(s) = 1 | − 91.4·2-s + 6.05e3·3-s − 2.44e4·4-s − 9.85e4·5-s − 5.53e5·6-s + 1.04e6·7-s + 5.22e6·8-s + 2.22e7·9-s + 9.00e6·10-s − 8.55e7·11-s − 1.47e8·12-s + 4.03e4·13-s − 9.59e7·14-s − 5.96e8·15-s + 3.22e8·16-s − 2.38e8·17-s − 2.03e9·18-s − 2.16e9·19-s + 2.40e9·20-s + 6.35e9·21-s + 7.81e9·22-s + 3.40e9·23-s + 3.16e10·24-s − 2.08e10·25-s − 3.69e6·26-s + 4.78e10·27-s − 2.56e10·28-s + ⋯ |
| L(s) = 1 | − 0.504·2-s + 1.59·3-s − 0.745·4-s − 0.564·5-s − 0.806·6-s + 0.481·7-s + 0.881·8-s + 1.55·9-s + 0.284·10-s − 1.32·11-s − 1.18·12-s + 0.000178·13-s − 0.243·14-s − 0.901·15-s + 0.300·16-s − 0.141·17-s − 0.783·18-s − 0.556·19-s + 0.420·20-s + 0.769·21-s + 0.668·22-s + 0.208·23-s + 1.40·24-s − 0.681·25-s − 9.01e − 5·26-s + 0.880·27-s − 0.359·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 23 | \( 1 - 3.40e9T \) |
| good | 2 | \( 1 + 91.4T + 3.27e4T^{2} \) |
| 3 | \( 1 - 6.05e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 9.85e4T + 3.05e10T^{2} \) |
| 7 | \( 1 - 1.04e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 8.55e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 4.03e4T + 5.11e16T^{2} \) |
| 17 | \( 1 + 2.38e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 2.16e9T + 1.51e19T^{2} \) |
| 29 | \( 1 - 5.67e10T + 8.62e21T^{2} \) |
| 31 | \( 1 + 1.97e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 6.04e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 2.51e11T + 1.55e24T^{2} \) |
| 43 | \( 1 - 1.45e12T + 3.17e24T^{2} \) |
| 47 | \( 1 + 3.92e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.33e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.87e13T + 3.65e26T^{2} \) |
| 61 | \( 1 + 1.26e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.97e12T + 2.46e27T^{2} \) |
| 71 | \( 1 - 3.79e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 6.27e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.36e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 3.19e14T + 6.11e28T^{2} \) |
| 89 | \( 1 - 7.15e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.34e15T + 6.33e29T^{2} \) |
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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
−13.79718388753845520666316745228, −12.81250588614645104448438316870, −10.65195590273018449609399130874, −9.324776204136707128862753125699, −8.239804831315158914299705147417, −7.66147779794152224213102698460, −4.76914364386756169851700788679, −3.42170155639912285786202304782, −1.87913357379212262994799827582, 0,
1.87913357379212262994799827582, 3.42170155639912285786202304782, 4.76914364386756169851700788679, 7.66147779794152224213102698460, 8.239804831315158914299705147417, 9.324776204136707128862753125699, 10.65195590273018449609399130874, 12.81250588614645104448438316870, 13.79718388753845520666316745228
