| L(s) = 1 | + 2-s − 1.49·3-s + 4-s + 1.25·5-s − 1.49·6-s − 0.926·7-s + 8-s − 0.775·9-s + 1.25·10-s − 3.21·11-s − 1.49·12-s + 0.860·13-s − 0.926·14-s − 1.87·15-s + 16-s + 5.37·17-s − 0.775·18-s − 0.0479·19-s + 1.25·20-s + 1.38·21-s − 3.21·22-s − 0.825·23-s − 1.49·24-s − 3.41·25-s + 0.860·26-s + 5.63·27-s − 0.926·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.861·3-s + 0.5·4-s + 0.563·5-s − 0.608·6-s − 0.350·7-s + 0.353·8-s − 0.258·9-s + 0.398·10-s − 0.969·11-s − 0.430·12-s + 0.238·13-s − 0.247·14-s − 0.484·15-s + 0.250·16-s + 1.30·17-s − 0.182·18-s − 0.0109·19-s + 0.281·20-s + 0.301·21-s − 0.685·22-s − 0.172·23-s − 0.304·24-s − 0.682·25-s + 0.168·26-s + 1.08·27-s − 0.175·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3001 | \( 1+O(T) \) |
| good | 3 | \( 1 + 1.49T + 3T^{2} \) |
| 5 | \( 1 - 1.25T + 5T^{2} \) |
| 7 | \( 1 + 0.926T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 13 | \( 1 - 0.860T + 13T^{2} \) |
| 17 | \( 1 - 5.37T + 17T^{2} \) |
| 19 | \( 1 + 0.0479T + 19T^{2} \) |
| 23 | \( 1 + 0.825T + 23T^{2} \) |
| 29 | \( 1 + 8.73T + 29T^{2} \) |
| 31 | \( 1 - 4.59T + 31T^{2} \) |
| 37 | \( 1 - 4.81T + 37T^{2} \) |
| 41 | \( 1 + 0.558T + 41T^{2} \) |
| 43 | \( 1 - 3.23T + 43T^{2} \) |
| 47 | \( 1 + 8.74T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 - 1.04T + 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 1.36T + 67T^{2} \) |
| 71 | \( 1 + 0.724T + 71T^{2} \) |
| 73 | \( 1 + 0.635T + 73T^{2} \) |
| 79 | \( 1 - 13.6T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 9.45T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.75110509807554655964736152504, −6.68301025986292161628301148068, −6.10785507921395455263470794085, −5.49719253366250395221193992777, −5.19278108398433036211923525373, −4.13053454996844064337302491470, −3.22127425906904282294866017778, −2.49941291637710588061451353393, −1.36775006618056388396019560922, 0,
1.36775006618056388396019560922, 2.49941291637710588061451353393, 3.22127425906904282294866017778, 4.13053454996844064337302491470, 5.19278108398433036211923525373, 5.49719253366250395221193992777, 6.10785507921395455263470794085, 6.68301025986292161628301148068, 7.75110509807554655964736152504
