| L(s) = 1 | − 142.·2-s − 1.74e3·3-s − 1.24e4·4-s − 3.36e5·5-s + 2.48e5·6-s + 2.12e5·7-s + 6.44e6·8-s − 1.13e7·9-s + 4.79e7·10-s + 3.84e7·11-s + 2.17e7·12-s + 3.30e8·13-s − 3.02e7·14-s + 5.87e8·15-s − 5.08e8·16-s − 5.30e8·17-s + 1.61e9·18-s − 5.44e9·19-s + 4.20e9·20-s − 3.69e8·21-s − 5.48e9·22-s + 2.03e10·23-s − 1.12e10·24-s + 8.28e10·25-s − 4.70e10·26-s + 4.47e10·27-s − 2.64e9·28-s + ⋯ |
| L(s) = 1 | − 0.786·2-s − 0.460·3-s − 0.381·4-s − 1.92·5-s + 0.362·6-s + 0.0973·7-s + 1.08·8-s − 0.788·9-s + 1.51·10-s + 0.595·11-s + 0.175·12-s + 1.46·13-s − 0.0766·14-s + 0.887·15-s − 0.473·16-s − 0.313·17-s + 0.620·18-s − 1.39·19-s + 0.734·20-s − 0.0448·21-s − 0.468·22-s + 1.24·23-s − 0.500·24-s + 2.71·25-s − 1.14·26-s + 0.822·27-s − 0.0371·28-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 | \( 1 + 142.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 1.74e3T + 1.43e7T^{2} \) |
| 5 | \( 1 + 3.36e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 2.12e5T + 4.74e12T^{2} \) |
| 11 | \( 1 - 3.84e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.30e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 5.30e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 5.44e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.03e10T + 2.66e20T^{2} \) |
| 31 | \( 1 - 1.01e11T + 2.34e22T^{2} \) |
| 37 | \( 1 + 7.61e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.52e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 4.98e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 2.79e12T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.28e13T + 7.31e25T^{2} \) |
| 59 | \( 1 - 3.15e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 3.37e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 6.01e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.20e14T + 5.87e27T^{2} \) |
| 73 | \( 1 - 1.99e13T + 8.90e27T^{2} \) |
| 79 | \( 1 + 1.72e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 1.29e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 3.15e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.29e14T + 6.33e29T^{2} \) |
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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
−12.83061284907010281786094763618, −11.40253400698099750844478114228, −10.86675116115484874225287295010, −8.730059439092101368309496259474, −8.274331971312178016865783317244, −6.71307565023329390589930122484, −4.66293542021698158131763115416, −3.56927673394750451578964355395, −0.962903765496821830137756770757, 0,
0.962903765496821830137756770757, 3.56927673394750451578964355395, 4.66293542021698158131763115416, 6.71307565023329390589930122484, 8.274331971312178016865783317244, 8.730059439092101368309496259474, 10.86675116115484874225287295010, 11.40253400698099750844478114228, 12.83061284907010281786094763618
