| L(s) = 1 | − 0.386·2-s + 3-s − 1.85·4-s + 0.287·5-s − 0.386·6-s − 2.57·7-s + 1.48·8-s + 9-s − 0.111·10-s − 4.70·11-s − 1.85·12-s − 0.906·13-s + 0.994·14-s + 0.287·15-s + 3.12·16-s + 4.50·17-s − 0.386·18-s + 3.83·19-s − 0.532·20-s − 2.57·21-s + 1.82·22-s − 23-s + 1.48·24-s − 4.91·25-s + 0.350·26-s + 27-s + 4.75·28-s + ⋯ |
| L(s) = 1 | − 0.273·2-s + 0.577·3-s − 0.925·4-s + 0.128·5-s − 0.157·6-s − 0.971·7-s + 0.526·8-s + 0.333·9-s − 0.0352·10-s − 1.41·11-s − 0.534·12-s − 0.251·13-s + 0.265·14-s + 0.0743·15-s + 0.781·16-s + 1.09·17-s − 0.0911·18-s + 0.878·19-s − 0.119·20-s − 0.561·21-s + 0.388·22-s − 0.208·23-s + 0.303·24-s − 0.983·25-s + 0.0687·26-s + 0.192·27-s + 0.899·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.120533767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120533767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 0.386T + 2T^{2} \) |
| 5 | \( 1 - 0.287T + 5T^{2} \) |
| 7 | \( 1 + 2.57T + 7T^{2} \) |
| 11 | \( 1 + 4.70T + 11T^{2} \) |
| 13 | \( 1 + 0.906T + 13T^{2} \) |
| 17 | \( 1 - 4.50T + 17T^{2} \) |
| 19 | \( 1 - 3.83T + 19T^{2} \) |
| 31 | \( 1 + 0.585T + 31T^{2} \) |
| 37 | \( 1 + 3.38T + 37T^{2} \) |
| 41 | \( 1 - 11.5T + 41T^{2} \) |
| 43 | \( 1 - 5.57T + 43T^{2} \) |
| 47 | \( 1 + 0.820T + 47T^{2} \) |
| 53 | \( 1 - 6.67T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 8.33T + 61T^{2} \) |
| 67 | \( 1 - 15.2T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 + 8.09T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 + 9.72T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 + 2.21T + 97T^{2} \) |
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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
−9.317696890181346228519346784213, −8.327883765477390392908284432680, −7.77765694914880517479933337021, −7.10903836976298325504740240170, −5.71434300089242070305866172017, −5.29301338997366834733724149627, −4.06147896478022852661030927539, −3.31293081936333959583866080682, −2.34811822495159269496871450090, −0.70630208399595489062974121298,
0.70630208399595489062974121298, 2.34811822495159269496871450090, 3.31293081936333959583866080682, 4.06147896478022852661030927539, 5.29301338997366834733724149627, 5.71434300089242070305866172017, 7.10903836976298325504740240170, 7.77765694914880517479933337021, 8.327883765477390392908284432680, 9.317696890181346228519346784213
