L(s) = 1 | − 0.0830·2-s − 1.99·4-s − 2.25·5-s − 3.20·7-s + 0.331·8-s + 0.186·10-s + 0.218·11-s − 4.17·13-s + 0.266·14-s + 3.95·16-s + 4.68·17-s + 3.43·19-s + 4.48·20-s − 0.0181·22-s + 3.99·23-s + 0.0635·25-s + 0.347·26-s + 6.39·28-s + 0.712·29-s + 1.04·31-s − 0.992·32-s − 0.389·34-s + 7.21·35-s + 1.70·37-s − 0.285·38-s − 0.746·40-s − 3.18·41-s + ⋯ |
L(s) = 1 | − 0.0587·2-s − 0.996·4-s − 1.00·5-s − 1.21·7-s + 0.117·8-s + 0.0590·10-s + 0.0657·11-s − 1.15·13-s + 0.0711·14-s + 0.989·16-s + 1.13·17-s + 0.789·19-s + 1.00·20-s − 0.00386·22-s + 0.833·23-s + 0.0127·25-s + 0.0680·26-s + 1.20·28-s + 0.132·29-s + 0.188·31-s − 0.175·32-s − 0.0667·34-s + 1.21·35-s + 0.281·37-s − 0.0463·38-s − 0.117·40-s − 0.497·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4527 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 2 | \( 1 + 0.0830T + 2T^{2} \) |
| 5 | \( 1 + 2.25T + 5T^{2} \) |
| 7 | \( 1 + 3.20T + 7T^{2} \) |
| 11 | \( 1 - 0.218T + 11T^{2} \) |
| 13 | \( 1 + 4.17T + 13T^{2} \) |
| 17 | \( 1 - 4.68T + 17T^{2} \) |
| 19 | \( 1 - 3.43T + 19T^{2} \) |
| 23 | \( 1 - 3.99T + 23T^{2} \) |
| 29 | \( 1 - 0.712T + 29T^{2} \) |
| 31 | \( 1 - 1.04T + 31T^{2} \) |
| 37 | \( 1 - 1.70T + 37T^{2} \) |
| 41 | \( 1 + 3.18T + 41T^{2} \) |
| 43 | \( 1 + 6.35T + 43T^{2} \) |
| 47 | \( 1 - 3.87T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 - 2.63T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.40T + 71T^{2} \) |
| 73 | \( 1 + 2.78T + 73T^{2} \) |
| 79 | \( 1 - 5.16T + 79T^{2} \) |
| 83 | \( 1 + 1.83T + 83T^{2} \) |
| 89 | \( 1 - 8.08T + 89T^{2} \) |
| 97 | \( 1 + 12.9T + 97T^{2} \) |
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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
−7.936661132592410164450837663634, −7.36399196904207677137869454223, −6.68859727212062869489859186577, −5.54317413373195281163716865205, −5.04618870514939929624497400416, −4.04607944864111320031031954687, −3.48324769595595052331724462640, −2.75096985839676928698666654709, −0.962948782632566206931793051727, 0,
0.962948782632566206931793051727, 2.75096985839676928698666654709, 3.48324769595595052331724462640, 4.04607944864111320031031954687, 5.04618870514939929624497400416, 5.54317413373195281163716865205, 6.68859727212062869489859186577, 7.36399196904207677137869454223, 7.936661132592410164450837663634