| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s + 9-s − 10-s − 12-s − 2·13-s + 2·14-s + 15-s + 16-s − 6·17-s + 18-s − 20-s − 2·21-s + 6·23-s − 24-s + 25-s − 2·26-s − 27-s + 2·28-s + 4·29-s + 30-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s − 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.534·14-s + 0.258·15-s + 1/4·16-s − 1.45·17-s + 0.235·18-s − 0.223·20-s − 0.436·21-s + 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.377·28-s + 0.742·29-s + 0.182·30-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.457563733\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.457563733\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 2 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−16.34354129465694, −15.87672574440883, −15.40786934430931, −14.72701134060816, −14.42131626329995, −13.53339245158761, −13.11674941350205, −12.42219784575820, −11.93285336220763, −11.35916451020652, −10.87833304004861, −10.45675953176315, −9.497438609302446, −8.767467256188853, −8.172719756180136, −7.282331913686833, −6.942564697946708, −6.232957196003574, −5.358216688559009, −4.794548506616312, −4.414412770011866, −3.523587114738642, −2.595077735860410, −1.800866135356497, −0.6716349815713708,
0.6716349815713708, 1.800866135356497, 2.595077735860410, 3.523587114738642, 4.414412770011866, 4.794548506616312, 5.358216688559009, 6.232957196003574, 6.942564697946708, 7.282331913686833, 8.172719756180136, 8.767467256188853, 9.497438609302446, 10.45675953176315, 10.87833304004861, 11.35916451020652, 11.93285336220763, 12.42219784575820, 13.11674941350205, 13.53339245158761, 14.42131626329995, 14.72701134060816, 15.40786934430931, 15.87672574440883, 16.34354129465694
