| L(s) = 1 | − 301.·2-s + 779.·3-s + 5.83e4·4-s − 2.35e5·6-s + 2.08e6·7-s − 7.73e6·8-s − 1.37e7·9-s − 7.71e7·11-s + 4.54e7·12-s + 1.32e8·13-s − 6.30e8·14-s + 4.21e8·16-s − 1.10e9·17-s + 4.14e9·18-s + 6.23e9·19-s + 1.62e9·21-s + 2.32e10·22-s + 2.49e10·23-s − 6.02e9·24-s − 3.99e10·26-s − 2.18e10·27-s + 1.21e11·28-s − 1.45e11·29-s + 4.01e10·31-s + 1.26e11·32-s − 6.01e10·33-s + 3.32e11·34-s + ⋯ |
| L(s) = 1 | − 1.66·2-s + 0.205·3-s + 1.78·4-s − 0.343·6-s + 0.958·7-s − 1.30·8-s − 0.957·9-s − 1.19·11-s + 0.366·12-s + 0.585·13-s − 1.59·14-s + 0.392·16-s − 0.651·17-s + 1.59·18-s + 1.60·19-s + 0.197·21-s + 1.99·22-s + 1.52·23-s − 0.268·24-s − 0.976·26-s − 0.402·27-s + 1.70·28-s − 1.56·29-s + 0.262·31-s + 0.648·32-s − 0.245·33-s + 1.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| good | 2 | \( 1 + 301.T + 3.27e4T^{2} \) |
| 3 | \( 1 - 779.T + 1.43e7T^{2} \) |
| 7 | \( 1 - 2.08e6T + 4.74e12T^{2} \) |
| 11 | \( 1 + 7.71e7T + 4.17e15T^{2} \) |
| 13 | \( 1 - 1.32e8T + 5.11e16T^{2} \) |
| 17 | \( 1 + 1.10e9T + 2.86e18T^{2} \) |
| 19 | \( 1 - 6.23e9T + 1.51e19T^{2} \) |
| 23 | \( 1 - 2.49e10T + 2.66e20T^{2} \) |
| 29 | \( 1 + 1.45e11T + 8.62e21T^{2} \) |
| 31 | \( 1 - 4.01e10T + 2.34e22T^{2} \) |
| 37 | \( 1 + 5.28e11T + 3.33e23T^{2} \) |
| 41 | \( 1 - 6.88e10T + 1.55e24T^{2} \) |
| 43 | \( 1 - 7.67e11T + 3.17e24T^{2} \) |
| 47 | \( 1 + 8.92e11T + 1.20e25T^{2} \) |
| 53 | \( 1 - 5.86e11T + 7.31e25T^{2} \) |
| 59 | \( 1 + 3.16e10T + 3.65e26T^{2} \) |
| 61 | \( 1 - 5.85e12T + 6.02e26T^{2} \) |
| 67 | \( 1 + 3.95e13T + 2.46e27T^{2} \) |
| 71 | \( 1 + 7.83e13T + 5.87e27T^{2} \) |
| 73 | \( 1 + 8.73e13T + 8.90e27T^{2} \) |
| 79 | \( 1 - 3.01e13T + 2.91e28T^{2} \) |
| 83 | \( 1 - 1.08e14T + 6.11e28T^{2} \) |
| 89 | \( 1 + 5.54e14T + 1.74e29T^{2} \) |
| 97 | \( 1 + 1.50e15T + 6.33e29T^{2} \) |
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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
−13.54033179947022412844610254554, −11.45494082862230615996187373734, −10.79130057307620394590584454272, −9.247125293141008090214004568061, −8.291230833966331299816345343700, −7.33450031024339869871177169617, −5.35407139247273341161340953671, −2.79743208068420612842878167010, −1.40296978256534162734038835886, 0,
1.40296978256534162734038835886, 2.79743208068420612842878167010, 5.35407139247273341161340953671, 7.33450031024339869871177169617, 8.291230833966331299816345343700, 9.247125293141008090214004568061, 10.79130057307620394590584454272, 11.45494082862230615996187373734, 13.54033179947022412844610254554
