| L(s) = 1 | − 9.50·2-s + 3.61e3·3-s − 3.26e4·4-s + 1.15e5·5-s − 3.43e4·6-s − 7.03e5·7-s + 6.22e5·8-s − 1.31e6·9-s − 1.09e6·10-s + 8.71e7·11-s − 1.17e8·12-s − 2.63e8·13-s + 6.68e6·14-s + 4.15e8·15-s + 1.06e9·16-s − 5.59e8·17-s + 1.25e7·18-s − 3.51e9·19-s − 3.76e9·20-s − 2.53e9·21-s − 8.28e8·22-s − 7.03e8·23-s + 2.24e9·24-s − 1.72e10·25-s + 2.50e9·26-s − 5.65e10·27-s + 2.29e10·28-s + ⋯ |
| L(s) = 1 | − 0.0525·2-s + 0.953·3-s − 0.997·4-s + 0.659·5-s − 0.0500·6-s − 0.322·7-s + 0.104·8-s − 0.0917·9-s − 0.0346·10-s + 1.34·11-s − 0.950·12-s − 1.16·13-s + 0.0169·14-s + 0.628·15-s + 0.991·16-s − 0.330·17-s + 0.00481·18-s − 0.902·19-s − 0.657·20-s − 0.307·21-s − 0.0708·22-s − 0.0430·23-s + 0.0999·24-s − 0.565·25-s + 0.0611·26-s − 1.04·27-s + 0.321·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 29 | \( 1 - 1.72e10T \) |
| good | 2 | \( 1 + 9.50T + 3.27e4T^{2} \) |
| 3 | \( 1 - 3.61e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 1.15e5T + 3.05e10T^{2} \) |
| 7 | \( 1 + 7.03e5T + 4.74e12T^{2} \) |
| 11 | \( 1 - 8.71e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 2.63e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 5.59e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 3.51e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 7.03e8T + 2.66e20T^{2} \) |
| 31 | \( 1 - 2.82e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 8.30e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.50e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 8.78e10T + 3.17e24T^{2} \) |
| 47 | \( 1 + 6.08e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 6.92e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 3.87e12T + 3.65e26T^{2} \) |
| 61 | \( 1 + 3.92e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 1.16e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 1.66e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 1.53e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.56e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.70e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 2.46e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.33e15T + 6.33e29T^{2} \) |
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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
−13.44494493712356819402798483868, −12.08529747536335801090389572266, −9.980683569613641756492084287665, −9.220366587528243313226004827161, −8.228534169478886367327340948494, −6.41398179399097721946827152955, −4.69559334394491599401302068348, −3.31441119273810636627488455760, −1.81470900349375909593623166596, 0,
1.81470900349375909593623166596, 3.31441119273810636627488455760, 4.69559334394491599401302068348, 6.41398179399097721946827152955, 8.228534169478886367327340948494, 9.220366587528243313226004827161, 9.980683569613641756492084287665, 12.08529747536335801090389572266, 13.44494493712356819402798483868
