L(s) = 1 | − 1.65·2-s − 0.430·3-s + 0.745·4-s + 5-s + 0.712·6-s + 7-s + 2.07·8-s − 2.81·9-s − 1.65·10-s − 0.615·11-s − 0.320·12-s − 2.52·13-s − 1.65·14-s − 0.430·15-s − 4.93·16-s − 3.21·17-s + 4.66·18-s − 0.349·19-s + 0.745·20-s − 0.430·21-s + 1.02·22-s + 5.68·23-s − 0.894·24-s + 25-s + 4.19·26-s + 2.50·27-s + 0.745·28-s + ⋯ |
L(s) = 1 | − 1.17·2-s − 0.248·3-s + 0.372·4-s + 0.447·5-s + 0.290·6-s + 0.377·7-s + 0.735·8-s − 0.938·9-s − 0.523·10-s − 0.185·11-s − 0.0925·12-s − 0.701·13-s − 0.442·14-s − 0.111·15-s − 1.23·16-s − 0.778·17-s + 1.09·18-s − 0.0802·19-s + 0.166·20-s − 0.0938·21-s + 0.217·22-s + 1.18·23-s − 0.182·24-s + 0.200·25-s + 0.821·26-s + 0.481·27-s + 0.140·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 229 | \( 1 + T \) |
good | 2 | \( 1 + 1.65T + 2T^{2} \) |
| 3 | \( 1 + 0.430T + 3T^{2} \) |
| 11 | \( 1 + 0.615T + 11T^{2} \) |
| 13 | \( 1 + 2.52T + 13T^{2} \) |
| 17 | \( 1 + 3.21T + 17T^{2} \) |
| 19 | \( 1 + 0.349T + 19T^{2} \) |
| 23 | \( 1 - 5.68T + 23T^{2} \) |
| 29 | \( 1 + 0.533T + 29T^{2} \) |
| 31 | \( 1 + 2.73T + 31T^{2} \) |
| 37 | \( 1 + 1.95T + 37T^{2} \) |
| 41 | \( 1 - 7.80T + 41T^{2} \) |
| 43 | \( 1 - 6.03T + 43T^{2} \) |
| 47 | \( 1 + 12.9T + 47T^{2} \) |
| 53 | \( 1 + 6.11T + 53T^{2} \) |
| 59 | \( 1 - 14.4T + 59T^{2} \) |
| 61 | \( 1 - 6.67T + 61T^{2} \) |
| 67 | \( 1 - 2.61T + 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 10.7T + 73T^{2} \) |
| 79 | \( 1 + 5.34T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 2.60T + 89T^{2} \) |
| 97 | \( 1 - 7.53T + 97T^{2} \) |
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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
−7.63028956015838603334605484890, −6.95036397689590242219352983625, −6.29961357454947548092670524861, −5.23420411775045319988618814814, −4.97716958324160359176818922607, −3.93498665289958895564810604754, −2.71597901136917016420716724456, −2.09712023341380297840779881225, −1.00738076849592371991780334094, 0,
1.00738076849592371991780334094, 2.09712023341380297840779881225, 2.71597901136917016420716724456, 3.93498665289958895564810604754, 4.97716958324160359176818922607, 5.23420411775045319988618814814, 6.29961357454947548092670524861, 6.95036397689590242219352983625, 7.63028956015838603334605484890