| L(s) = 1 | + 2.73·3-s − 5-s + 4.47·7-s + 4.45·9-s + 4.94·11-s + 2.80·13-s − 2.73·15-s + 4.39·17-s + 19-s + 12.2·21-s + 4.47·23-s + 25-s + 3.97·27-s − 10.3·29-s − 3.88·31-s + 13.4·33-s − 4.47·35-s − 4.61·37-s + 7.65·39-s + 1.09·41-s + 3.85·43-s − 4.45·45-s − 11.3·47-s + 13.0·49-s + 12.0·51-s + 13.6·53-s − 4.94·55-s + ⋯ |
| L(s) = 1 | + 1.57·3-s − 0.447·5-s + 1.69·7-s + 1.48·9-s + 1.49·11-s + 0.778·13-s − 0.704·15-s + 1.06·17-s + 0.229·19-s + 2.66·21-s + 0.932·23-s + 0.200·25-s + 0.764·27-s − 1.91·29-s − 0.698·31-s + 2.34·33-s − 0.756·35-s − 0.758·37-s + 1.22·39-s + 0.170·41-s + 0.587·43-s − 0.664·45-s − 1.65·47-s + 1.85·49-s + 1.68·51-s + 1.86·53-s − 0.666·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.072885852\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.072885852\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 4.47T + 7T^{2} \) |
| 11 | \( 1 - 4.94T + 11T^{2} \) |
| 13 | \( 1 - 2.80T + 13T^{2} \) |
| 17 | \( 1 - 4.39T + 17T^{2} \) |
| 23 | \( 1 - 4.47T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 + 3.88T + 31T^{2} \) |
| 37 | \( 1 + 4.61T + 37T^{2} \) |
| 41 | \( 1 - 1.09T + 41T^{2} \) |
| 43 | \( 1 - 3.85T + 43T^{2} \) |
| 47 | \( 1 + 11.3T + 47T^{2} \) |
| 53 | \( 1 - 13.6T + 53T^{2} \) |
| 59 | \( 1 + 7.09T + 59T^{2} \) |
| 61 | \( 1 + 5.09T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 + 0.909T + 71T^{2} \) |
| 73 | \( 1 + 0.0281T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 15.5T + 83T^{2} \) |
| 89 | \( 1 + 2.51T + 89T^{2} \) |
| 97 | \( 1 - 9.13T + 97T^{2} \) |
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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
−8.167787329157054242139356293708, −7.44070253264739446696625637565, −7.15360979044333275957186609493, −5.86778326630594932866590048478, −5.06169068596804607001459015997, −4.03856993630651616023211271516, −3.76429162928713597445824610134, −2.88295472783922896126823232165, −1.59822197317502181450713672768, −1.40697883848180223508111990211,
1.40697883848180223508111990211, 1.59822197317502181450713672768, 2.88295472783922896126823232165, 3.76429162928713597445824610134, 4.03856993630651616023211271516, 5.06169068596804607001459015997, 5.86778326630594932866590048478, 7.15360979044333275957186609493, 7.44070253264739446696625637565, 8.167787329157054242139356293708
