| L(s) = 1 | − 2·3-s − 7-s + 9-s + 11-s + 2·13-s − 2·17-s − 4·19-s + 2·21-s − 5·25-s + 4·27-s − 29-s − 8·31-s − 2·33-s − 6·37-s − 4·39-s − 6·41-s − 12·43-s − 12·47-s + 49-s + 4·51-s − 10·53-s + 8·57-s + 4·59-s − 63-s − 4·67-s + 14·73-s + 10·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.554·13-s − 0.485·17-s − 0.917·19-s + 0.436·21-s − 25-s + 0.769·27-s − 0.185·29-s − 1.43·31-s − 0.348·33-s − 0.986·37-s − 0.640·39-s − 0.937·41-s − 1.82·43-s − 1.75·47-s + 1/7·49-s + 0.560·51-s − 1.37·53-s + 1.05·57-s + 0.520·59-s − 0.125·63-s − 0.488·67-s + 1.63·73-s + 1.15·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 35728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 35728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−15.47105934460097, −15.04672132827008, −14.41234399545148, −13.81395682989015, −13.18070651629184, −12.84763208740215, −12.21548600478555, −11.67286788591569, −11.27317507750490, −10.80707126486344, −10.26851704420299, −9.676600543622508, −9.094829174289863, −8.393655388104067, −8.037555002489500, −6.935639447249975, −6.691490837285983, −6.205000703201365, −5.498578731639725, −5.110560568808272, −4.332269405079234, −3.674293726298590, −3.092033106902644, −1.950140602573316, −1.455800271345212, 0, 0,
1.455800271345212, 1.950140602573316, 3.092033106902644, 3.674293726298590, 4.332269405079234, 5.110560568808272, 5.498578731639725, 6.205000703201365, 6.691490837285983, 6.935639447249975, 8.037555002489500, 8.393655388104067, 9.094829174289863, 9.676600543622508, 10.26851704420299, 10.80707126486344, 11.27317507750490, 11.67286788591569, 12.21548600478555, 12.84763208740215, 13.18070651629184, 13.81395682989015, 14.41234399545148, 15.04672132827008, 15.47105934460097
