| L(s) = 1 | − 107.·2-s + 5.77e3·3-s − 2.11e4·4-s + 2.51e5·5-s − 6.23e5·6-s + 1.51e6·7-s + 5.81e6·8-s + 1.89e7·9-s − 2.71e7·10-s − 2.83e7·11-s − 1.21e8·12-s + 3.32e8·13-s − 1.63e8·14-s + 1.45e9·15-s + 6.43e7·16-s + 4.11e8·17-s − 2.05e9·18-s + 1.55e9·19-s − 5.30e9·20-s + 8.72e9·21-s + 3.06e9·22-s − 2.19e10·23-s + 3.35e10·24-s + 3.25e10·25-s − 3.58e10·26-s + 2.68e10·27-s − 3.19e10·28-s + ⋯ |
| L(s) = 1 | − 0.596·2-s + 1.52·3-s − 0.644·4-s + 1.43·5-s − 0.908·6-s + 0.693·7-s + 0.980·8-s + 1.32·9-s − 0.857·10-s − 0.438·11-s − 0.982·12-s + 1.46·13-s − 0.413·14-s + 2.19·15-s + 0.0599·16-s + 0.242·17-s − 0.789·18-s + 0.399·19-s − 0.926·20-s + 1.05·21-s + 0.261·22-s − 1.34·23-s + 1.49·24-s + 1.06·25-s − 0.875·26-s + 0.493·27-s − 0.447·28-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
| \(L(8)\) |
\(\approx\) |
\(3.341234385\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.341234385\) |
| \(L(\frac{17}{2})\) |
|
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 | \( 1 + 107.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 5.77e3T + 1.43e7T^{2} \) |
| 5 | \( 1 - 2.51e5T + 3.05e10T^{2} \) |
| 7 | \( 1 - 1.51e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 2.83e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 3.32e8T + 5.11e16T^{2} \) |
| 17 | \( 1 - 4.11e8T + 2.86e18T^{2} \) |
| 19 | \( 1 - 1.55e9T + 1.51e19T^{2} \) |
| 23 | \( 1 + 2.19e10T + 2.66e20T^{2} \) |
| 31 | \( 1 + 8.77e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 6.53e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 1.66e12T + 1.55e24T^{2} \) |
| 43 | \( 1 - 3.50e12T + 3.17e24T^{2} \) |
| 47 | \( 1 - 3.84e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 4.86e12T + 7.31e25T^{2} \) |
| 59 | \( 1 - 1.55e13T + 3.65e26T^{2} \) |
| 61 | \( 1 - 3.51e13T + 6.02e26T^{2} \) |
| 67 | \( 1 + 7.33e13T + 2.46e27T^{2} \) |
| 71 | \( 1 - 1.24e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 3.34e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 2.73e14T + 2.91e28T^{2} \) |
| 83 | \( 1 + 9.85e12T + 6.11e28T^{2} \) |
| 89 | \( 1 - 6.37e13T + 1.74e29T^{2} \) |
| 97 | \( 1 + 6.14e14T + 6.33e29T^{2} \) |
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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
−13.86845057429720120784593634988, −13.08623387262511373411999627433, −10.58050643194332093143700254459, −9.512301911901190531915066238976, −8.704322078654143008173768510846, −7.76303112834945938777643803480, −5.62691012795206335788617849399, −3.87986516878761229125914893600, −2.20545936285730893322790446568, −1.26204312219445591456479619165,
1.26204312219445591456479619165, 2.20545936285730893322790446568, 3.87986516878761229125914893600, 5.62691012795206335788617849399, 7.76303112834945938777643803480, 8.704322078654143008173768510846, 9.512301911901190531915066238976, 10.58050643194332093143700254459, 13.08623387262511373411999627433, 13.86845057429720120784593634988
