L(s) = 1 | − 2-s − 3-s + 4-s − 4·5-s + 6-s + 7-s − 8-s + 9-s + 4·10-s + 2·11-s − 12-s + 2·13-s − 14-s + 4·15-s + 16-s + 2·17-s − 18-s − 2·19-s − 4·20-s − 21-s − 2·22-s + 23-s + 24-s + 11·25-s − 2·26-s − 27-s + 28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s − 0.894·20-s − 0.218·21-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 11/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.590145272834824356845533988063, −8.517749678089344145487901033794, −8.057806690325987987133750230050, −7.17223625973566331940704514276, −6.47795508864117195965745614559, −5.18705288465943249189670817747, −4.12890850837478174257592345990, −3.32637588427684486740674767745, −1.41523106223574871080970468543, 0,
1.41523106223574871080970468543, 3.32637588427684486740674767745, 4.12890850837478174257592345990, 5.18705288465943249189670817747, 6.47795508864117195965745614559, 7.17223625973566331940704514276, 8.057806690325987987133750230050, 8.517749678089344145487901033794, 9.590145272834824356845533988063