| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s − 13-s + 16-s − 2·17-s + 2·20-s − 3·23-s − 25-s + 26-s − 29-s − 7·31-s − 32-s + 2·34-s + 4·37-s − 2·40-s + 5·41-s + 43-s + 3·46-s − 2·47-s + 50-s − 52-s − 10·53-s + 58-s + 3·59-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s − 0.277·13-s + 1/4·16-s − 0.485·17-s + 0.447·20-s − 0.625·23-s − 1/5·25-s + 0.196·26-s − 0.185·29-s − 1.25·31-s − 0.176·32-s + 0.342·34-s + 0.657·37-s − 0.316·40-s + 0.780·41-s + 0.152·43-s + 0.442·46-s − 0.291·47-s + 0.141·50-s − 0.138·52-s − 1.37·53-s + 0.131·58-s + 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 106722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 106722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.130263817\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.130263817\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 5 T + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−13.70356987295806, −13.16003259506869, −12.73560700107710, −12.22927923865932, −11.64747344552725, −11.04404608205312, −10.81186488033762, −10.11189411651874, −9.649030171388816, −9.400832574644245, −8.834024773341899, −8.286300409523317, −7.658303933983088, −7.334376221672170, −6.578325968966390, −6.187895432199949, −5.664889971427224, −5.168320728854313, −4.389061353838910, −3.839575911084540, −3.024835225040769, −2.434549582206762, −1.871561235202663, −1.371123253055036, −0.3591284946031212,
0.3591284946031212, 1.371123253055036, 1.871561235202663, 2.434549582206762, 3.024835225040769, 3.839575911084540, 4.389061353838910, 5.168320728854313, 5.664889971427224, 6.187895432199949, 6.578325968966390, 7.334376221672170, 7.658303933983088, 8.286300409523317, 8.834024773341899, 9.400832574644245, 9.649030171388816, 10.11189411651874, 10.81186488033762, 11.04404608205312, 11.64747344552725, 12.22927923865932, 12.73560700107710, 13.16003259506869, 13.70356987295806
