| L(s) = 1 | − 2-s + 0.304·3-s + 4-s − 0.0530·5-s − 0.304·6-s + 2.56·7-s − 8-s − 2.90·9-s + 0.0530·10-s + 3.82·11-s + 0.304·12-s − 2.14·13-s − 2.56·14-s − 0.0161·15-s + 16-s − 6.77·17-s + 2.90·18-s − 0.247·19-s − 0.0530·20-s + 0.780·21-s − 3.82·22-s + 7.18·23-s − 0.304·24-s − 4.99·25-s + 2.14·26-s − 1.79·27-s + 2.56·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.175·3-s + 0.5·4-s − 0.0237·5-s − 0.124·6-s + 0.968·7-s − 0.353·8-s − 0.969·9-s + 0.0167·10-s + 1.15·11-s + 0.0879·12-s − 0.593·13-s − 0.684·14-s − 0.00417·15-s + 0.250·16-s − 1.64·17-s + 0.685·18-s − 0.0568·19-s − 0.0118·20-s + 0.170·21-s − 0.815·22-s + 1.49·23-s − 0.0621·24-s − 0.999·25-s + 0.419·26-s − 0.346·27-s + 0.484·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 0.304T + 3T^{2} \) |
| 5 | \( 1 + 0.0530T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 - 3.82T + 11T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 + 6.77T + 17T^{2} \) |
| 19 | \( 1 + 0.247T + 19T^{2} \) |
| 23 | \( 1 - 7.18T + 23T^{2} \) |
| 29 | \( 1 - 3.18T + 29T^{2} \) |
| 31 | \( 1 + 9.94T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 + 3.68T + 41T^{2} \) |
| 43 | \( 1 + 4.51T + 43T^{2} \) |
| 47 | \( 1 + 8.22T + 47T^{2} \) |
| 53 | \( 1 - 0.273T + 53T^{2} \) |
| 59 | \( 1 + 8.23T + 59T^{2} \) |
| 61 | \( 1 - 7.42T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 + 12.8T + 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 1.61T + 79T^{2} \) |
| 83 | \( 1 - 6.66T + 83T^{2} \) |
| 89 | \( 1 + 4.99T + 89T^{2} \) |
| 97 | \( 1 + 0.590T + 97T^{2} \) |
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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
−8.186701171625576285985728531015, −7.47979011548154000956677769429, −6.73170491456993434525576933704, −6.03389046518783376167744113476, −5.04271026937659054483874169488, −4.32467321050720691201582562679, −3.22742273145714186675610292751, −2.26414480527922354019808990004, −1.45750359203771531019701337837, 0,
1.45750359203771531019701337837, 2.26414480527922354019808990004, 3.22742273145714186675610292751, 4.32467321050720691201582562679, 5.04271026937659054483874169488, 6.03389046518783376167744113476, 6.73170491456993434525576933704, 7.47979011548154000956677769429, 8.186701171625576285985728531015
