| L(s) = 1 | + 0.618·2-s − 1.61·4-s − 3.85·5-s − 2.23·7-s − 2.23·8-s − 2.38·10-s − 1.38·11-s − 0.236·13-s − 1.38·14-s + 1.85·16-s − 4.38·17-s + 4.85·19-s + 6.23·20-s − 0.854·22-s + 1.23·23-s + 9.85·25-s − 0.145·26-s + 3.61·28-s + 10.0·31-s + 5.61·32-s − 2.70·34-s + 8.61·35-s − 4.70·37-s + 3.00·38-s + 8.61·40-s + 3.85·41-s + 7.23·43-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.809·4-s − 1.72·5-s − 0.845·7-s − 0.790·8-s − 0.753·10-s − 0.416·11-s − 0.0654·13-s − 0.369·14-s + 0.463·16-s − 1.06·17-s + 1.11·19-s + 1.39·20-s − 0.182·22-s + 0.257·23-s + 1.97·25-s − 0.0286·26-s + 0.683·28-s + 1.81·31-s + 0.993·32-s − 0.464·34-s + 1.45·35-s − 0.774·37-s + 0.486·38-s + 1.36·40-s + 0.601·41-s + 1.10·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7569 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 29 | \( 1 \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 5 | \( 1 + 3.85T + 5T^{2} \) |
| 7 | \( 1 + 2.23T + 7T^{2} \) |
| 11 | \( 1 + 1.38T + 11T^{2} \) |
| 13 | \( 1 + 0.236T + 13T^{2} \) |
| 17 | \( 1 + 4.38T + 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 23 | \( 1 - 1.23T + 23T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 4.70T + 37T^{2} \) |
| 41 | \( 1 - 3.85T + 41T^{2} \) |
| 43 | \( 1 - 7.23T + 43T^{2} \) |
| 47 | \( 1 + 7T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 6.09T + 59T^{2} \) |
| 61 | \( 1 - 0.618T + 61T^{2} \) |
| 67 | \( 1 + 1.52T + 67T^{2} \) |
| 71 | \( 1 + 10.4T + 71T^{2} \) |
| 73 | \( 1 - 13.7T + 73T^{2} \) |
| 79 | \( 1 - 6.09T + 79T^{2} \) |
| 83 | \( 1 - 9.94T + 83T^{2} \) |
| 89 | \( 1 - 4.70T + 89T^{2} \) |
| 97 | \( 1 + 3.56T + 97T^{2} \) |
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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
−7.66888809270511735506656688390, −6.81826373381767942757709600017, −6.19861821317192020058503048207, −5.13648996936056286004662386940, −4.62360037753779704083530271442, −3.95325514022545368171845629529, −3.29663496028882584131834961666, −2.72593181406276068344884815786, −0.839107629013873499571080072480, 0,
0.839107629013873499571080072480, 2.72593181406276068344884815786, 3.29663496028882584131834961666, 3.95325514022545368171845629529, 4.62360037753779704083530271442, 5.13648996936056286004662386940, 6.19861821317192020058503048207, 6.81826373381767942757709600017, 7.66888809270511735506656688390
