| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s − 3·13-s − 14-s + 15-s + 16-s + 8·17-s − 18-s − 20-s − 21-s − 22-s + 24-s + 25-s + 3·26-s − 27-s + 28-s − 6·29-s − 30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s + 0.301·11-s − 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.258·15-s + 1/4·16-s + 1.94·17-s − 0.235·18-s − 0.223·20-s − 0.218·21-s − 0.213·22-s + 0.204·24-s + 1/5·25-s + 0.588·26-s − 0.192·27-s + 0.188·28-s − 1.11·29-s − 0.182·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 174570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 7 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.32278736915357, −12.77488648481845, −12.22045320205613, −11.96040730875581, −11.61007179388998, −11.04868858207988, −10.59919496647722, −9.979718347979221, −9.774811595964685, −9.281121716450685, −8.509074095362977, −8.107522919437170, −7.706521891370084, −7.209477600639405, −6.773421462180300, −6.178458894329489, −5.550427505383002, −5.124606043702606, −4.674844170086645, −3.823955592425937, −3.406001429055082, −2.779318857298374, −1.916443405786824, −1.416027654795930, −0.7357499879234213, 0,
0.7357499879234213, 1.416027654795930, 1.916443405786824, 2.779318857298374, 3.406001429055082, 3.823955592425937, 4.674844170086645, 5.124606043702606, 5.550427505383002, 6.178458894329489, 6.773421462180300, 7.209477600639405, 7.706521891370084, 8.107522919437170, 8.509074095362977, 9.281121716450685, 9.774811595964685, 9.979718347979221, 10.59919496647722, 11.04868858207988, 11.61007179388998, 11.96040730875581, 12.22045320205613, 12.77488648481845, 13.32278736915357
