| L(s) = 1 | + 0.436·3-s + 0.299·5-s − 2.80·9-s − 5.18·11-s + 1.26·13-s + 0.130·15-s − 1.95·17-s − 19-s − 1.49·23-s − 4.91·25-s − 2.53·27-s − 0.530·29-s − 1.94·31-s − 2.26·33-s + 4.19·37-s + 0.550·39-s + 2·41-s + 5.51·43-s − 0.841·45-s + 8.54·47-s − 0.855·51-s − 11.7·53-s − 1.55·55-s − 0.436·57-s + 12.8·59-s + 5.93·61-s + 0.378·65-s + ⋯ |
| L(s) = 1 | + 0.252·3-s + 0.134·5-s − 0.936·9-s − 1.56·11-s + 0.349·13-s + 0.0337·15-s − 0.475·17-s − 0.229·19-s − 0.310·23-s − 0.982·25-s − 0.488·27-s − 0.0984·29-s − 0.349·31-s − 0.394·33-s + 0.689·37-s + 0.0882·39-s + 0.312·41-s + 0.841·43-s − 0.125·45-s + 1.24·47-s − 0.119·51-s − 1.61·53-s − 0.209·55-s − 0.0578·57-s + 1.66·59-s + 0.760·61-s + 0.0468·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.316233586\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.316233586\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.436T + 3T^{2} \) |
| 5 | \( 1 - 0.299T + 5T^{2} \) |
| 11 | \( 1 + 5.18T + 11T^{2} \) |
| 13 | \( 1 - 1.26T + 13T^{2} \) |
| 17 | \( 1 + 1.95T + 17T^{2} \) |
| 23 | \( 1 + 1.49T + 23T^{2} \) |
| 29 | \( 1 + 0.530T + 29T^{2} \) |
| 31 | \( 1 + 1.94T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 5.51T + 43T^{2} \) |
| 47 | \( 1 - 8.54T + 47T^{2} \) |
| 53 | \( 1 + 11.7T + 53T^{2} \) |
| 59 | \( 1 - 12.8T + 59T^{2} \) |
| 61 | \( 1 - 5.93T + 61T^{2} \) |
| 67 | \( 1 - 5.50T + 67T^{2} \) |
| 71 | \( 1 - 2.28T + 71T^{2} \) |
| 73 | \( 1 - 4.69T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 1.46T + 83T^{2} \) |
| 89 | \( 1 - 3.85T + 89T^{2} \) |
| 97 | \( 1 - 9.69T + 97T^{2} \) |
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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
−7.983840766591749370097826467275, −7.38131013234753079953958340758, −6.39324096660359490226684295278, −5.72490630634561692692103624073, −5.25249740368999988721852415775, −4.28522472751684021972061930304, −3.47866123063882592819711684670, −2.55444892489361382595997187067, −2.10753831087071319223293367197, −0.53411161804527664603932231875,
0.53411161804527664603932231875, 2.10753831087071319223293367197, 2.55444892489361382595997187067, 3.47866123063882592819711684670, 4.28522472751684021972061930304, 5.25249740368999988721852415775, 5.72490630634561692692103624073, 6.39324096660359490226684295278, 7.38131013234753079953958340758, 7.983840766591749370097826467275
