| L(s) = 1 | + 0.618·2-s − 0.381·3-s − 1.61·4-s − 1.61·5-s − 0.236·6-s + 7-s − 2.23·8-s − 2.85·9-s − 1.00·10-s − 2.23·11-s + 0.618·12-s − 3·13-s + 0.618·14-s + 0.618·15-s + 1.85·16-s + 1.47·17-s − 1.76·18-s − 4.85·19-s + 2.61·20-s − 0.381·21-s − 1.38·22-s + 0.381·23-s + 0.854·24-s − 2.38·25-s − 1.85·26-s + 2.23·27-s − 1.61·28-s + ⋯ |
| L(s) = 1 | + 0.437·2-s − 0.220·3-s − 0.809·4-s − 0.723·5-s − 0.0963·6-s + 0.377·7-s − 0.790·8-s − 0.951·9-s − 0.316·10-s − 0.674·11-s + 0.178·12-s − 0.832·13-s + 0.165·14-s + 0.159·15-s + 0.463·16-s + 0.357·17-s − 0.415·18-s − 1.11·19-s + 0.585·20-s − 0.0833·21-s − 0.294·22-s + 0.0796·23-s + 0.174·24-s − 0.476·25-s − 0.363·26-s + 0.430·27-s − 0.305·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 - 0.618T + 2T^{2} \) |
| 3 | \( 1 + 0.381T + 3T^{2} \) |
| 5 | \( 1 + 1.61T + 5T^{2} \) |
| 11 | \( 1 + 2.23T + 11T^{2} \) |
| 13 | \( 1 + 3T + 13T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 + 4.85T + 19T^{2} \) |
| 23 | \( 1 - 0.381T + 23T^{2} \) |
| 29 | \( 1 - 7.85T + 29T^{2} \) |
| 31 | \( 1 + 9.61T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 43 | \( 1 - 7.94T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 + 14.0T + 53T^{2} \) |
| 59 | \( 1 - 7.61T + 59T^{2} \) |
| 61 | \( 1 + 11T + 61T^{2} \) |
| 67 | \( 1 - 4.38T + 67T^{2} \) |
| 71 | \( 1 - 0.236T + 71T^{2} \) |
| 73 | \( 1 + 15T + 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 14.5T + 89T^{2} \) |
| 97 | \( 1 + 5.14T + 97T^{2} \) |
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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
−11.49051920929202154611488628707, −10.56553151263908002424453819058, −9.370888851144348331161595704357, −8.368925400919503796376739483112, −7.66323616568412668951613256629, −6.07548332091169831833081580045, −5.09345513963198659437801107628, −4.19630942537214621639294326356, −2.82674548134373629608035344615, 0,
2.82674548134373629608035344615, 4.19630942537214621639294326356, 5.09345513963198659437801107628, 6.07548332091169831833081580045, 7.66323616568412668951613256629, 8.368925400919503796376739483112, 9.370888851144348331161595704357, 10.56553151263908002424453819058, 11.49051920929202154611488628707
