| L(s) = 1 | + 3-s − 0.473·5-s + 4.55·7-s + 9-s + 3.49·11-s − 0.0840·13-s − 0.473·15-s − 3.61·17-s + 3.61·19-s + 4.55·21-s + 2.82·23-s − 4.77·25-s + 27-s + 7.30·29-s − 0.557·31-s + 3.49·33-s − 2.15·35-s + 6.20·37-s − 0.0840·39-s − 9.27·41-s + 2.27·43-s − 0.473·45-s − 2.82·47-s + 13.7·49-s − 3.61·51-s + 0.697·53-s − 1.65·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.211·5-s + 1.72·7-s + 0.333·9-s + 1.05·11-s − 0.0233·13-s − 0.122·15-s − 0.877·17-s + 0.829·19-s + 0.994·21-s + 0.589·23-s − 0.955·25-s + 0.192·27-s + 1.35·29-s − 0.100·31-s + 0.608·33-s − 0.364·35-s + 1.01·37-s − 0.0134·39-s − 1.44·41-s + 0.347·43-s − 0.0706·45-s − 0.412·47-s + 1.96·49-s − 0.506·51-s + 0.0958·53-s − 0.223·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.034707355\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.034707355\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| good | 5 | \( 1 + 0.473T + 5T^{2} \) |
| 7 | \( 1 - 4.55T + 7T^{2} \) |
| 11 | \( 1 - 3.49T + 11T^{2} \) |
| 13 | \( 1 + 0.0840T + 13T^{2} \) |
| 17 | \( 1 + 3.61T + 17T^{2} \) |
| 19 | \( 1 - 3.61T + 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 - 7.30T + 29T^{2} \) |
| 31 | \( 1 + 0.557T + 31T^{2} \) |
| 37 | \( 1 - 6.20T + 37T^{2} \) |
| 41 | \( 1 + 9.27T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 - 0.697T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 3.85T + 61T^{2} \) |
| 67 | \( 1 + 5.33T + 67T^{2} \) |
| 71 | \( 1 + 9.11T + 71T^{2} \) |
| 73 | \( 1 + 0.541T + 73T^{2} \) |
| 79 | \( 1 - 10.9T + 79T^{2} \) |
| 83 | \( 1 + 15.0T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 - 4.31T + 97T^{2} \) |
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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
−8.597990846808329581752590527873, −8.081098253245847289376613544499, −7.33864589463273855708209282839, −6.62046219484732688216654238249, −5.53980266290598741579356042350, −4.61583065950290170096756665315, −4.17520230710405257669018800579, −3.05089411415083314282680706107, −1.94999727565317841272604993228, −1.15424370902459972779351263586,
1.15424370902459972779351263586, 1.94999727565317841272604993228, 3.05089411415083314282680706107, 4.17520230710405257669018800579, 4.61583065950290170096756665315, 5.53980266290598741579356042350, 6.62046219484732688216654238249, 7.33864589463273855708209282839, 8.081098253245847289376613544499, 8.597990846808329581752590527873
