| L(s) = 1 | − 13.8·2-s + 79.8·3-s + 64.3·4-s − 355.·5-s − 1.10e3·6-s + 343·7-s + 882.·8-s + 4.18e3·9-s + 4.93e3·10-s − 3.97e3·11-s + 5.13e3·12-s − 2.19e3·13-s − 4.75e3·14-s − 2.83e4·15-s − 2.04e4·16-s + 3.20e4·17-s − 5.80e4·18-s − 3.46e4·19-s − 2.29e4·20-s + 2.73e4·21-s + 5.51e4·22-s − 3.52e4·23-s + 7.04e4·24-s + 4.83e4·25-s + 3.04e4·26-s + 1.59e5·27-s + 2.20e4·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 1.70·3-s + 0.503·4-s − 1.27·5-s − 2.09·6-s + 0.377·7-s + 0.609·8-s + 1.91·9-s + 1.56·10-s − 0.901·11-s + 0.858·12-s − 0.277·13-s − 0.463·14-s − 2.17·15-s − 1.24·16-s + 1.58·17-s − 2.34·18-s − 1.15·19-s − 0.640·20-s + 0.645·21-s + 1.10·22-s − 0.603·23-s + 1.04·24-s + 0.619·25-s + 0.340·26-s + 1.55·27-s + 0.190·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 13.8T + 128T^{2} \) |
| 3 | \( 1 - 79.8T + 2.18e3T^{2} \) |
| 5 | \( 1 + 355.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 3.97e3T + 1.94e7T^{2} \) |
| 17 | \( 1 - 3.20e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.46e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 3.52e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.61e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 3.15e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.47e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.87e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.31e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.09e6T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.10e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.03e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.10e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.31e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.26e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 4.07e5T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.59e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.19e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.73e5T + 8.07e13T^{2} \) |
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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
−12.08186049586631986906771536207, −10.57313147305697761646588936935, −9.705565609986521738440430165052, −8.407140531253178250913579359742, −8.061092284932148524636953709698, −7.35201737992491333176015863373, −4.49694805464871564065538603051, −3.22282454670445935353522658596, −1.72815867989601915891903134667, 0,
1.72815867989601915891903134667, 3.22282454670445935353522658596, 4.49694805464871564065538603051, 7.35201737992491333176015863373, 8.061092284932148524636953709698, 8.407140531253178250913579359742, 9.705565609986521738440430165052, 10.57313147305697761646588936935, 12.08186049586631986906771536207
