| L(s) = 1 | − 0.164·3-s − 1.38·5-s − 1.87·7-s − 2.97·9-s − 0.536·11-s − 3.43·13-s + 0.227·15-s − 5.29·17-s − 2.51·19-s + 0.308·21-s − 1.08·23-s − 3.09·25-s + 0.985·27-s + 1.95·29-s + 10.5·31-s + 0.0884·33-s + 2.58·35-s − 6.13·37-s + 0.566·39-s + 6.36·41-s + 0.652·43-s + 4.10·45-s − 10.2·47-s − 3.50·49-s + 0.872·51-s − 1.31·53-s + 0.741·55-s + ⋯ |
| L(s) = 1 | − 0.0952·3-s − 0.617·5-s − 0.706·7-s − 0.990·9-s − 0.161·11-s − 0.952·13-s + 0.0588·15-s − 1.28·17-s − 0.577·19-s + 0.0672·21-s − 0.225·23-s − 0.618·25-s + 0.189·27-s + 0.362·29-s + 1.89·31-s + 0.0153·33-s + 0.436·35-s − 1.00·37-s + 0.0907·39-s + 0.993·41-s + 0.0995·43-s + 0.612·45-s − 1.50·47-s − 0.500·49-s + 0.122·51-s − 0.181·53-s + 0.0999·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5474683169\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5474683169\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 0.164T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 + 0.536T + 11T^{2} \) |
| 13 | \( 1 + 3.43T + 13T^{2} \) |
| 17 | \( 1 + 5.29T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 + 1.08T + 23T^{2} \) |
| 29 | \( 1 - 1.95T + 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 + 6.13T + 37T^{2} \) |
| 41 | \( 1 - 6.36T + 41T^{2} \) |
| 43 | \( 1 - 0.652T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 + 1.31T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 + 8.27T + 61T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 - 8.24T + 71T^{2} \) |
| 73 | \( 1 - 7.43T + 73T^{2} \) |
| 79 | \( 1 - 7.33T + 79T^{2} \) |
| 83 | \( 1 + 7.83T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−8.249187804585119464447272498141, −7.966575622128066275223077909643, −6.66872377341975989734544169677, −6.54973074096625118737053959936, −5.42508499504785303689845188334, −4.66162721536937318460164841951, −3.85709068189552022448814291059, −2.88724248669968195854544902518, −2.21812808458645933143628542712, −0.39442739359651450890787571390,
0.39442739359651450890787571390, 2.21812808458645933143628542712, 2.88724248669968195854544902518, 3.85709068189552022448814291059, 4.66162721536937318460164841951, 5.42508499504785303689845188334, 6.54973074096625118737053959936, 6.66872377341975989734544169677, 7.966575622128066275223077909643, 8.249187804585119464447272498141
