| L(s) = 1 | − 2·3-s − 2·7-s + 9-s − 4·11-s − 8·17-s + 19-s + 4·21-s − 6·23-s + 4·27-s − 2·29-s − 8·31-s + 8·33-s − 6·41-s + 10·43-s + 6·47-s − 3·49-s + 16·51-s − 2·57-s + 4·59-s − 6·61-s − 2·63-s + 2·67-s + 12·69-s + 16·71-s + 16·73-s + 8·77-s + 8·79-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.755·7-s + 1/3·9-s − 1.20·11-s − 1.94·17-s + 0.229·19-s + 0.872·21-s − 1.25·23-s + 0.769·27-s − 0.371·29-s − 1.43·31-s + 1.39·33-s − 0.937·41-s + 1.52·43-s + 0.875·47-s − 3/7·49-s + 2.24·51-s − 0.264·57-s + 0.520·59-s − 0.768·61-s − 0.251·63-s + 0.244·67-s + 1.44·69-s + 1.89·71-s + 1.87·73-s + 0.911·77-s + 0.900·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−15.47929651170066, −15.19310980728027, −14.09423968124454, −13.81888997523047, −13.08978905537363, −12.65022189292676, −12.34979366601201, −11.49495223106343, −11.12236013806829, −10.65062329788903, −10.23431007670267, −9.420719309784422, −9.060711803960938, −8.219340009593665, −7.697119659700912, −6.876549517452288, −6.546904461702442, −5.862835773586780, −5.412596348104517, −4.835673503628117, −4.100978694887558, −3.432434062405760, −2.454777730061812, −2.011130095683093, −0.5930138501410244, 0,
0.5930138501410244, 2.011130095683093, 2.454777730061812, 3.432434062405760, 4.100978694887558, 4.835673503628117, 5.412596348104517, 5.862835773586780, 6.546904461702442, 6.876549517452288, 7.697119659700912, 8.219340009593665, 9.060711803960938, 9.420719309784422, 10.23431007670267, 10.65062329788903, 11.12236013806829, 11.49495223106343, 12.34979366601201, 12.65022189292676, 13.08978905537363, 13.81888997523047, 14.09423968124454, 15.19310980728027, 15.47929651170066
