| L(s) = 1 | + 1.37·5-s − 7-s + 4.37·11-s + 2·13-s + 4.37·17-s − 5·19-s − 7.37·23-s − 3.11·25-s − 2.74·29-s − 2·31-s − 1.37·35-s + 2·37-s − 10.3·41-s − 9.11·43-s + 49-s − 2.74·53-s + 6·55-s + 7.11·59-s − 14.1·61-s + 2.74·65-s − 15.1·67-s + 10.1·71-s − 5.11·73-s − 4.37·77-s − 12.1·79-s − 5.48·83-s + 6·85-s + ⋯ |
| L(s) = 1 | + 0.613·5-s − 0.377·7-s + 1.31·11-s + 0.554·13-s + 1.06·17-s − 1.14·19-s − 1.53·23-s − 0.623·25-s − 0.509·29-s − 0.359·31-s − 0.231·35-s + 0.328·37-s − 1.61·41-s − 1.39·43-s + 0.142·49-s − 0.376·53-s + 0.809·55-s + 0.926·59-s − 1.80·61-s + 0.340·65-s − 1.84·67-s + 1.20·71-s − 0.598·73-s − 0.498·77-s − 1.36·79-s − 0.602·83-s + 0.650·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 1.37T + 5T^{2} \) |
| 11 | \( 1 - 4.37T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4.37T + 17T^{2} \) |
| 19 | \( 1 + 5T + 19T^{2} \) |
| 23 | \( 1 + 7.37T + 23T^{2} \) |
| 29 | \( 1 + 2.74T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 9.11T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 2.74T + 53T^{2} \) |
| 59 | \( 1 - 7.11T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 15.1T + 67T^{2} \) |
| 71 | \( 1 - 10.1T + 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 + 12.1T + 79T^{2} \) |
| 83 | \( 1 + 5.48T + 83T^{2} \) |
| 89 | \( 1 + 3.25T + 89T^{2} \) |
| 97 | \( 1 - 9.11T + 97T^{2} \) |
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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
−7.33904782818921849819659807796, −6.49012786480387222411372933903, −6.11430880883318722000610078856, −5.54950983348875706202560869292, −4.47326216120974344022097209619, −3.80099467177175591140117293803, −3.20087802776295477107327487739, −1.94077573619259854784712224247, −1.47798059349005779261588754731, 0,
1.47798059349005779261588754731, 1.94077573619259854784712224247, 3.20087802776295477107327487739, 3.80099467177175591140117293803, 4.47326216120974344022097209619, 5.54950983348875706202560869292, 6.11430880883318722000610078856, 6.49012786480387222411372933903, 7.33904782818921849819659807796
