| L(s) = 1 | + 1.73·2-s + 2.73·3-s + 0.999·4-s + 1.73·5-s + 4.73·6-s + 7-s − 1.73·8-s + 4.46·9-s + 2.99·10-s + 1.26·11-s + 2.73·12-s + 1.73·14-s + 4.73·15-s − 5·16-s − 7.73·17-s + 7.73·18-s + 2·19-s + 1.73·20-s + 2.73·21-s + 2.19·22-s + 4.73·23-s − 4.73·24-s − 2.00·25-s + 3.99·27-s + 0.999·28-s − 3·29-s + 8.19·30-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 1.57·3-s + 0.499·4-s + 0.774·5-s + 1.93·6-s + 0.377·7-s − 0.612·8-s + 1.48·9-s + 0.948·10-s + 0.382·11-s + 0.788·12-s + 0.462·14-s + 1.22·15-s − 1.25·16-s − 1.87·17-s + 1.82·18-s + 0.458·19-s + 0.387·20-s + 0.596·21-s + 0.468·22-s + 0.986·23-s − 0.965·24-s − 0.400·25-s + 0.769·27-s + 0.188·28-s − 0.557·29-s + 1.49·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.356228563\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.356228563\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 17 | \( 1 + 7.73T + 17T^{2} \) |
| 19 | \( 1 - 2T + 19T^{2} \) |
| 23 | \( 1 - 4.73T + 23T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 - 4.19T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + 5.19T + 41T^{2} \) |
| 43 | \( 1 + 0.196T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 9.92T + 53T^{2} \) |
| 59 | \( 1 - 7.26T + 59T^{2} \) |
| 61 | \( 1 + 4.80T + 61T^{2} \) |
| 67 | \( 1 + 6.19T + 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 3.19T + 73T^{2} \) |
| 79 | \( 1 - 16.1T + 79T^{2} \) |
| 83 | \( 1 + 2.19T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 6.39T + 97T^{2} \) |
| 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
−9.324299252313651527737534493362, −9.117789009467200424218151363662, −8.286577858241470359999842963459, −7.14258485507133526607743722122, −6.38594470424405032777692483958, −5.27951855114179207636657680047, −4.42485224867946784061644121196, −3.60165692141157611830664999333, −2.65069316426082859006632021710, −1.87763258882109013351049874304,
1.87763258882109013351049874304, 2.65069316426082859006632021710, 3.60165692141157611830664999333, 4.42485224867946784061644121196, 5.27951855114179207636657680047, 6.38594470424405032777692483958, 7.14258485507133526607743722122, 8.286577858241470359999842963459, 9.117789009467200424218151363662, 9.324299252313651527737534493362
