| L(s) = 1 | − 64·2-s − 388.·3-s + 4.09e3·4-s + 2.59e4·5-s + 2.48e4·6-s − 2.62e5·8-s − 1.44e6·9-s − 1.65e6·10-s − 2.25e5·11-s − 1.59e6·12-s + 2.48e7·13-s − 1.00e7·15-s + 1.67e7·16-s + 1.22e8·17-s + 9.23e7·18-s − 2.14e8·19-s + 1.06e8·20-s + 1.44e7·22-s − 9.81e8·23-s + 1.01e8·24-s − 5.49e8·25-s − 1.58e9·26-s + 1.18e9·27-s − 2.47e9·29-s + 6.44e8·30-s − 4.88e9·31-s − 1.07e9·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.307·3-s + 0.5·4-s + 0.741·5-s + 0.217·6-s − 0.353·8-s − 0.905·9-s − 0.524·10-s − 0.0383·11-s − 0.153·12-s + 1.42·13-s − 0.228·15-s + 0.250·16-s + 1.22·17-s + 0.640·18-s − 1.04·19-s + 0.370·20-s + 0.0271·22-s − 1.38·23-s + 0.108·24-s − 0.450·25-s − 1.00·26-s + 0.586·27-s − 0.772·29-s + 0.161·30-s − 0.988·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 64T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 388.T + 1.59e6T^{2} \) |
| 5 | \( 1 - 2.59e4T + 1.22e9T^{2} \) |
| 11 | \( 1 + 2.25e5T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.48e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.22e8T + 9.90e15T^{2} \) |
| 19 | \( 1 + 2.14e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 9.81e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.47e9T + 1.02e19T^{2} \) |
| 31 | \( 1 + 4.88e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 2.70e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 3.27e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 5.78e9T + 1.71e21T^{2} \) |
| 47 | \( 1 - 1.51e10T + 5.46e21T^{2} \) |
| 53 | \( 1 - 3.93e10T + 2.60e22T^{2} \) |
| 59 | \( 1 - 1.44e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 1.57e11T + 1.61e23T^{2} \) |
| 67 | \( 1 + 7.67e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.73e12T + 1.16e24T^{2} \) |
| 73 | \( 1 - 1.58e12T + 1.67e24T^{2} \) |
| 79 | \( 1 + 2.86e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 9.41e10T + 8.87e24T^{2} \) |
| 89 | \( 1 + 3.95e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 1.04e13T + 6.73e25T^{2} \) |
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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
−10.76868878813645000601935305711, −9.756502530510965633145642295099, −8.718097954286477363076493743812, −7.76672555865046380085322772990, −6.09150678743583654083064488366, −5.78112951488150930591153094295, −3.80148926708249528675945437083, −2.37643617659455829937685024457, −1.25491926689055955766880584681, 0,
1.25491926689055955766880584681, 2.37643617659455829937685024457, 3.80148926708249528675945437083, 5.78112951488150930591153094295, 6.09150678743583654083064488366, 7.76672555865046380085322772990, 8.718097954286477363076493743812, 9.756502530510965633145642295099, 10.76868878813645000601935305711
