L(s) = 1 | − 0.904·3-s − 1.47·5-s + 0.136·7-s − 2.18·9-s − 3.00·11-s − 1.49·13-s + 1.33·15-s + 2.83·17-s − 19-s − 0.123·21-s + 5.06·23-s − 2.83·25-s + 4.68·27-s + 5.50·29-s + 3.46·31-s + 2.71·33-s − 0.200·35-s + 6.80·37-s + 1.35·39-s + 4.08·41-s + 11.0·43-s + 3.21·45-s − 3.64·47-s − 6.98·49-s − 2.56·51-s + 2.72·53-s + 4.42·55-s + ⋯ |
L(s) = 1 | − 0.522·3-s − 0.658·5-s + 0.0514·7-s − 0.727·9-s − 0.906·11-s − 0.413·13-s + 0.343·15-s + 0.687·17-s − 0.229·19-s − 0.0268·21-s + 1.05·23-s − 0.566·25-s + 0.902·27-s + 1.02·29-s + 0.623·31-s + 0.473·33-s − 0.0338·35-s + 1.11·37-s + 0.216·39-s + 0.637·41-s + 1.68·43-s + 0.478·45-s − 0.531·47-s − 0.997·49-s − 0.359·51-s + 0.373·53-s + 0.596·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 19 | \( 1 + T \) |
| 79 | \( 1 - T \) |
good | 3 | \( 1 + 0.904T + 3T^{2} \) |
| 5 | \( 1 + 1.47T + 5T^{2} \) |
| 7 | \( 1 - 0.136T + 7T^{2} \) |
| 11 | \( 1 + 3.00T + 11T^{2} \) |
| 13 | \( 1 + 1.49T + 13T^{2} \) |
| 17 | \( 1 - 2.83T + 17T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 - 5.50T + 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.80T + 37T^{2} \) |
| 41 | \( 1 - 4.08T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 3.64T + 47T^{2} \) |
| 53 | \( 1 - 2.72T + 53T^{2} \) |
| 59 | \( 1 + 3.70T + 59T^{2} \) |
| 61 | \( 1 - 4.64T + 61T^{2} \) |
| 67 | \( 1 + 14.7T + 67T^{2} \) |
| 71 | \( 1 + 1.65T + 71T^{2} \) |
| 73 | \( 1 + 3.04T + 73T^{2} \) |
| 83 | \( 1 + 12.5T + 83T^{2} \) |
| 89 | \( 1 + 0.595T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−7.81253942086832141440605505671, −7.09492761815931532022711464747, −6.18234679168545517136905970810, −5.60504587033882593448121530481, −4.84570827060578917531988022972, −4.22150363222486721091191965258, −3.03802631636652903866231207922, −2.59723775788526212668157373312, −1.04411537883096492235789799743, 0,
1.04411537883096492235789799743, 2.59723775788526212668157373312, 3.03802631636652903866231207922, 4.22150363222486721091191965258, 4.84570827060578917531988022972, 5.60504587033882593448121530481, 6.18234679168545517136905970810, 7.09492761815931532022711464747, 7.81253942086832141440605505671