L(s) = 1 | − 2-s + 1.73·3-s + 1.73·5-s − 1.73·6-s + 7-s + 8-s + 1.99·9-s − 1.73·10-s − 11-s − 14-s + 2.99·15-s − 16-s − 1.73·17-s − 1.99·18-s + 1.73·21-s + 22-s − 2·23-s + 1.73·24-s + 1.99·25-s + 1.73·27-s + 29-s − 2.99·30-s − 1.73·33-s + 1.73·34-s + 1.73·35-s + 1.73·40-s + 1.73·41-s + ⋯ |
L(s) = 1 | − 2-s + 1.73·3-s + 1.73·5-s − 1.73·6-s + 7-s + 8-s + 1.99·9-s − 1.73·10-s − 11-s − 14-s + 2.99·15-s − 16-s − 1.73·17-s − 1.99·18-s + 1.73·21-s + 22-s − 2·23-s + 1.73·24-s + 1.99·25-s + 1.73·27-s + 29-s − 2.99·30-s − 1.73·33-s + 1.73·34-s + 1.73·35-s + 1.73·40-s + 1.73·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3311 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.782859634\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.782859634\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 2 | \( 1 + T + T^{2} \) |
| 3 | \( 1 - 1.73T + T^{2} \) |
| 5 | \( 1 - 1.73T + T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + 1.73T + T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 2T + T^{2} \) |
| 29 | \( 1 - T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.73T + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 1.73T + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.830394831869553802566133763004, −8.255777371708133237294004229608, −7.80086988507720216542135620637, −6.90176486483825017176334350034, −5.89376937023417299936277552583, −4.78559549623376918382131606876, −4.23847231206451009986118420675, −2.67679709496079848597541878789, −2.12159053728325003148277221466, −1.56879683563627832377873696796,
1.56879683563627832377873696796, 2.12159053728325003148277221466, 2.67679709496079848597541878789, 4.23847231206451009986118420675, 4.78559549623376918382131606876, 5.89376937023417299936277552583, 6.90176486483825017176334350034, 7.80086988507720216542135620637, 8.255777371708133237294004229608, 8.830394831869553802566133763004