| L(s) = 1 | + 2.99·3-s − 2.48·5-s + 3.43·7-s + 5.97·9-s + 4.32·11-s − 5.29·13-s − 7.43·15-s + 0.604·17-s + 4.40·19-s + 10.2·21-s − 1.89·23-s + 1.16·25-s + 8.90·27-s + 3.54·29-s + 0.110·31-s + 12.9·33-s − 8.51·35-s − 4.59·37-s − 15.8·39-s + 6.22·41-s + 9.99·43-s − 14.8·45-s + 5.22·47-s + 4.76·49-s + 1.81·51-s + 4.78·53-s − 10.7·55-s + ⋯ |
| L(s) = 1 | + 1.72·3-s − 1.11·5-s + 1.29·7-s + 1.99·9-s + 1.30·11-s − 1.46·13-s − 1.92·15-s + 0.146·17-s + 1.01·19-s + 2.24·21-s − 0.395·23-s + 0.233·25-s + 1.71·27-s + 0.658·29-s + 0.0197·31-s + 2.25·33-s − 1.43·35-s − 0.754·37-s − 2.53·39-s + 0.972·41-s + 1.52·43-s − 2.21·45-s + 0.762·47-s + 0.681·49-s + 0.253·51-s + 0.657·53-s − 1.44·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.270579265\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.270579265\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 + 2.48T + 5T^{2} \) |
| 7 | \( 1 - 3.43T + 7T^{2} \) |
| 11 | \( 1 - 4.32T + 11T^{2} \) |
| 13 | \( 1 + 5.29T + 13T^{2} \) |
| 17 | \( 1 - 0.604T + 17T^{2} \) |
| 19 | \( 1 - 4.40T + 19T^{2} \) |
| 23 | \( 1 + 1.89T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 - 0.110T + 31T^{2} \) |
| 37 | \( 1 + 4.59T + 37T^{2} \) |
| 41 | \( 1 - 6.22T + 41T^{2} \) |
| 43 | \( 1 - 9.99T + 43T^{2} \) |
| 47 | \( 1 - 5.22T + 47T^{2} \) |
| 53 | \( 1 - 4.78T + 53T^{2} \) |
| 59 | \( 1 + 14.9T + 59T^{2} \) |
| 61 | \( 1 - 6.74T + 61T^{2} \) |
| 67 | \( 1 - 8.85T + 67T^{2} \) |
| 71 | \( 1 + 0.223T + 71T^{2} \) |
| 73 | \( 1 + 0.173T + 73T^{2} \) |
| 79 | \( 1 + 5.92T + 79T^{2} \) |
| 83 | \( 1 + 3.15T + 83T^{2} \) |
| 89 | \( 1 + 5.16T + 89T^{2} \) |
| 97 | \( 1 - 1.14T + 97T^{2} \) |
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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
−7.980377133246596231100076742042, −7.32131749708349174915184045337, −7.01457123179554722758600747254, −5.62232784050358624655461962275, −4.56944379298003370437262077754, −4.25006561139753336350922653935, −3.52647464599939058514325019593, −2.70700632601030011420379354472, −1.92448915397199836849300127555, −0.997464353405990092696821313859,
0.997464353405990092696821313859, 1.92448915397199836849300127555, 2.70700632601030011420379354472, 3.52647464599939058514325019593, 4.25006561139753336350922653935, 4.56944379298003370437262077754, 5.62232784050358624655461962275, 7.01457123179554722758600747254, 7.32131749708349174915184045337, 7.980377133246596231100076742042
