L(s) = 1 | + 1.83·3-s + 2·5-s − 1.53·7-s + 0.363·9-s − 6.05·11-s − 5.50·13-s + 3.66·15-s + 1.11·17-s + 2.87·19-s − 2.81·21-s − 6.59·23-s − 25-s − 4.83·27-s + 3.97·29-s + 1.23·31-s − 11.0·33-s − 3.07·35-s + 11.1·37-s − 10.1·39-s + 2.55·41-s − 11.0·43-s + 0.726·45-s − 8.14·47-s − 4.63·49-s + 2.04·51-s − 6.62·53-s − 12.1·55-s + ⋯ |
L(s) = 1 | + 1.05·3-s + 0.894·5-s − 0.580·7-s + 0.121·9-s − 1.82·11-s − 1.52·13-s + 0.946·15-s + 0.270·17-s + 0.658·19-s − 0.614·21-s − 1.37·23-s − 0.200·25-s − 0.930·27-s + 0.737·29-s + 0.221·31-s − 1.93·33-s − 0.519·35-s + 1.83·37-s − 1.61·39-s + 0.398·41-s − 1.68·43-s + 0.108·45-s − 1.18·47-s − 0.662·49-s + 0.286·51-s − 0.910·53-s − 1.63·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 167 | \( 1 - T \) |
good | 3 | \( 1 - 1.83T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 + 6.05T + 11T^{2} \) |
| 13 | \( 1 + 5.50T + 13T^{2} \) |
| 17 | \( 1 - 1.11T + 17T^{2} \) |
| 19 | \( 1 - 2.87T + 19T^{2} \) |
| 23 | \( 1 + 6.59T + 23T^{2} \) |
| 29 | \( 1 - 3.97T + 29T^{2} \) |
| 31 | \( 1 - 1.23T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 - 2.55T + 41T^{2} \) |
| 43 | \( 1 + 11.0T + 43T^{2} \) |
| 47 | \( 1 + 8.14T + 47T^{2} \) |
| 53 | \( 1 + 6.62T + 53T^{2} \) |
| 59 | \( 1 - 12.3T + 59T^{2} \) |
| 61 | \( 1 + 0.918T + 61T^{2} \) |
| 67 | \( 1 + 12.9T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 5.47T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 4.26T + 89T^{2} \) |
| 97 | \( 1 + 16.8T + 97T^{2} \) |
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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
−8.183375216764664393942423316099, −8.009901057257333555491397395094, −7.12545289496664213549960899702, −6.06061885955686726588149951344, −5.39569187803141055049246628625, −4.56712423201397618784157281776, −3.19654766131830629242163619295, −2.66331227441011915528014045676, −1.99747302304752598877130531434, 0,
1.99747302304752598877130531434, 2.66331227441011915528014045676, 3.19654766131830629242163619295, 4.56712423201397618784157281776, 5.39569187803141055049246628625, 6.06061885955686726588149951344, 7.12545289496664213549960899702, 8.009901057257333555491397395094, 8.183375216764664393942423316099