| L(s) = 1 | − 2-s + 3-s + 4-s − 5-s − 6-s + 7-s − 8-s + 9-s + 10-s + 12-s + 13-s − 14-s − 15-s + 16-s − 6·17-s − 18-s + 19-s − 20-s + 21-s − 24-s + 25-s − 26-s + 27-s + 28-s + 6·29-s + 30-s − 4·31-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 + 0.288·12-s + 0.277·13-s − 0.267·14-s − 0.258·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s + 0.229·19-s − 0.223·20-s + 0.218·21-s − 0.204·24-s + 1/5·25-s − 0.196·26-s + 0.192·27-s + 0.188·28-s + 1.11·29-s + 0.182·30-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51870 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−14.71528882022883, −14.38249763290644, −13.74384194702688, −13.16880192780379, −12.76111731075319, −12.05226829781911, −11.55679213652881, −11.07846200264149, −10.62870373625871, −10.06317602892849, −9.390330712063779, −8.958279458951882, −8.459330876109583, −8.098189466434214, −7.374584476759068, −7.058254914088917, −6.337540734103240, −5.821749797974669, −4.811681741748454, −4.469159316751511, −3.710138374418765, −3.034667796850677, −2.395255210232274, −1.737515275693739, −0.9602794677125935, 0,
0.9602794677125935, 1.737515275693739, 2.395255210232274, 3.034667796850677, 3.710138374418765, 4.469159316751511, 4.811681741748454, 5.821749797974669, 6.337540734103240, 7.058254914088917, 7.374584476759068, 8.098189466434214, 8.459330876109583, 8.958279458951882, 9.390330712063779, 10.06317602892849, 10.62870373625871, 11.07846200264149, 11.55679213652881, 12.05226829781911, 12.76111731075319, 13.16880192780379, 13.74384194702688, 14.38249763290644, 14.71528882022883
