| L(s) = 1 | + 2.65·3-s − 2.96·5-s + 7-s + 4.07·9-s + 4.80·11-s − 13-s − 7.87·15-s + 0.268·17-s + 6.50·19-s + 2.65·21-s − 2.30·23-s + 3.76·25-s + 2.85·27-s + 6.54·29-s − 4.03·31-s + 12.7·33-s − 2.96·35-s + 2.51·37-s − 2.65·39-s + 4.46·41-s − 10.6·43-s − 12.0·45-s + 8.63·47-s + 49-s + 0.714·51-s − 3.37·53-s − 14.2·55-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 1.32·5-s + 0.377·7-s + 1.35·9-s + 1.44·11-s − 0.277·13-s − 2.03·15-s + 0.0651·17-s + 1.49·19-s + 0.580·21-s − 0.480·23-s + 0.753·25-s + 0.548·27-s + 1.21·29-s − 0.724·31-s + 2.22·33-s − 0.500·35-s + 0.413·37-s − 0.425·39-s + 0.697·41-s − 1.63·43-s − 1.79·45-s + 1.25·47-s + 0.142·49-s + 0.100·51-s − 0.463·53-s − 1.91·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.637373796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.637373796\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.65T + 3T^{2} \) |
| 5 | \( 1 + 2.96T + 5T^{2} \) |
| 11 | \( 1 - 4.80T + 11T^{2} \) |
| 17 | \( 1 - 0.268T + 17T^{2} \) |
| 19 | \( 1 - 6.50T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 - 6.54T + 29T^{2} \) |
| 31 | \( 1 + 4.03T + 31T^{2} \) |
| 37 | \( 1 - 2.51T + 37T^{2} \) |
| 41 | \( 1 - 4.46T + 41T^{2} \) |
| 43 | \( 1 + 10.6T + 43T^{2} \) |
| 47 | \( 1 - 8.63T + 47T^{2} \) |
| 53 | \( 1 + 3.37T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 9.13T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 16.2T + 71T^{2} \) |
| 73 | \( 1 - 6.03T + 73T^{2} \) |
| 79 | \( 1 - 5.76T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 - 6.64T + 89T^{2} \) |
| 97 | \( 1 + 2.74T + 97T^{2} \) |
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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
−9.310496845304703701680019102292, −8.636254393239763273641049475693, −7.944915593650875323616210680087, −7.43599032907286762731458098668, −6.59996766351663245059525794545, −5.08902595807367296765603056477, −3.94655793272870004978591299734, −3.65916027286259779864518485332, −2.55514922754463282122681960400, −1.19643797830723953806787828872,
1.19643797830723953806787828872, 2.55514922754463282122681960400, 3.65916027286259779864518485332, 3.94655793272870004978591299734, 5.08902595807367296765603056477, 6.59996766351663245059525794545, 7.43599032907286762731458098668, 7.944915593650875323616210680087, 8.636254393239763273641049475693, 9.310496845304703701680019102292
