| L(s) = 1 | + 320.·2-s − 2.70e3·3-s + 6.96e4·4-s + 5.74e4·5-s − 8.67e5·6-s − 3.02e6·7-s + 1.18e7·8-s − 7.00e6·9-s + 1.83e7·10-s + 7.24e7·11-s − 1.88e8·12-s − 3.69e8·13-s − 9.67e8·14-s − 1.55e8·15-s + 1.49e9·16-s − 1.67e8·17-s − 2.24e9·18-s − 4.86e9·19-s + 4.00e9·20-s + 8.19e9·21-s + 2.31e10·22-s + 1.40e10·23-s − 3.19e10·24-s − 2.72e10·25-s − 1.18e11·26-s + 5.78e10·27-s − 2.10e11·28-s + ⋯ |
| L(s) = 1 | + 1.76·2-s − 0.715·3-s + 2.12·4-s + 0.329·5-s − 1.26·6-s − 1.38·7-s + 1.98·8-s − 0.488·9-s + 0.581·10-s + 1.12·11-s − 1.52·12-s − 1.63·13-s − 2.45·14-s − 0.235·15-s + 1.39·16-s − 0.0990·17-s − 0.863·18-s − 1.24·19-s + 0.699·20-s + 0.992·21-s + 1.98·22-s + 0.858·23-s − 1.42·24-s − 0.891·25-s − 2.88·26-s + 1.06·27-s − 2.94·28-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 | \( 1 - 320.T + 3.27e4T^{2} \) |
| 3 | \( 1 + 2.70e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 5.74e4T + 3.05e10T^{2} \) |
| 7 | \( 1 + 3.02e6T + 4.74e12T^{2} \) |
| 11 | \( 1 - 7.24e7T + 4.17e15T^{2} \) |
| 13 | \( 1 + 3.69e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.67e8T + 2.86e18T^{2} \) |
| 19 | \( 1 + 4.86e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 1.40e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 2.48e11T + 2.34e22T^{2} \) |
| 37 | \( 1 - 5.41e11T + 3.33e23T^{2} \) |
| 41 | \( 1 + 1.06e12T + 1.55e24T^{2} \) |
| 43 | \( 1 + 6.53e11T + 3.17e24T^{2} \) |
| 47 | \( 1 - 3.74e11T + 1.20e25T^{2} \) |
| 53 | \( 1 + 1.31e13T + 7.31e25T^{2} \) |
| 59 | \( 1 + 1.64e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 2.76e13T + 6.02e26T^{2} \) |
| 67 | \( 1 - 6.77e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 7.57e13T + 5.87e27T^{2} \) |
| 73 | \( 1 - 6.78e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 1.40e14T + 2.91e28T^{2} \) |
| 83 | \( 1 - 2.18e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 2.10e14T + 1.74e29T^{2} \) |
| 97 | \( 1 - 1.22e14T + 6.33e29T^{2} \) |
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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
−12.89897794205247450486845345183, −12.25174631786357744987744358031, −11.11247822566473398150746953311, −9.542566734802223732387651682072, −6.83904361795382072383202959159, −6.15628903239399328148237334995, −4.97762161328303325539495680797, −3.58178433773773480656034200600, −2.30881760202251729314703306555, 0,
2.30881760202251729314703306555, 3.58178433773773480656034200600, 4.97762161328303325539495680797, 6.15628903239399328148237334995, 6.83904361795382072383202959159, 9.542566734802223732387651682072, 11.11247822566473398150746953311, 12.25174631786357744987744358031, 12.89897794205247450486845345183
