| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 3·7-s − 8-s + 9-s + 10-s − 6·11-s − 12-s + 13-s − 3·14-s + 15-s + 16-s + 3·17-s − 18-s − 2·19-s − 20-s − 3·21-s + 6·22-s − 4·23-s + 24-s + 25-s − 26-s − 27-s + 3·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 1.13·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.80·11-s − 0.288·12-s + 0.277·13-s − 0.801·14-s + 0.258·15-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 0.458·19-s − 0.223·20-s − 0.654·21-s + 1.27·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.196·26-s − 0.192·27-s + 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 374790 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 \) |
| good | 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.56200959784333, −12.09835519851406, −11.82447850043244, −11.18843229708246, −10.86120357185915, −10.53215387382389, −10.16877537266650, −9.688461569448379, −8.949540067702053, −8.520111817996753, −8.092258258789995, −7.657305119703811, −7.526499749437059, −6.842915273871454, −6.187854330359576, −5.731141545906120, −5.306349343243761, −4.741351485456857, −4.463848204508755, −3.598824138427724, −3.124651941841670, −2.455586316320011, −1.809462023533802, −1.444392310919112, −0.5506497028418276, 0,
0.5506497028418276, 1.444392310919112, 1.809462023533802, 2.455586316320011, 3.124651941841670, 3.598824138427724, 4.463848204508755, 4.741351485456857, 5.306349343243761, 5.731141545906120, 6.187854330359576, 6.842915273871454, 7.526499749437059, 7.657305119703811, 8.092258258789995, 8.520111817996753, 8.949540067702053, 9.688461569448379, 10.16877537266650, 10.53215387382389, 10.86120357185915, 11.18843229708246, 11.82447850043244, 12.09835519851406, 12.56200959784333
