| L(s) = 1 | − 0.153·2-s − 4.07·3-s − 31.9·4-s + 25·5-s + 0.627·6-s − 123.·7-s + 9.83·8-s − 226.·9-s − 3.84·10-s + 121·11-s + 130.·12-s + 468.·13-s + 18.9·14-s − 101.·15-s + 1.02e3·16-s + 238.·17-s + 34.8·18-s + 361·19-s − 799.·20-s + 502.·21-s − 18.6·22-s − 4.22e3·23-s − 40.1·24-s + 625·25-s − 72.0·26-s + 1.91e3·27-s + 3.93e3·28-s + ⋯ |
| L(s) = 1 | − 0.0271·2-s − 0.261·3-s − 0.999·4-s + 0.447·5-s + 0.00711·6-s − 0.950·7-s + 0.0543·8-s − 0.931·9-s − 0.0121·10-s + 0.301·11-s + 0.261·12-s + 0.768·13-s + 0.0258·14-s − 0.117·15-s + 0.997·16-s + 0.199·17-s + 0.0253·18-s + 0.229·19-s − 0.446·20-s + 0.248·21-s − 0.00819·22-s − 1.66·23-s − 0.0142·24-s + 0.200·25-s − 0.0208·26-s + 0.505·27-s + 0.949·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6063721487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6063721487\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 - 361T \) |
| good | 2 | \( 1 + 0.153T + 32T^{2} \) |
| 3 | \( 1 + 4.07T + 243T^{2} \) |
| 7 | \( 1 + 123.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 468.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 238.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 4.22e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 773.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 8.89e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 9.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.93e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 6.87e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.19e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.00e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.59e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.09e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 9.19e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.48e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.71e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.12e4T + 8.58e9T^{2} \) |
| 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
−9.092818003430239067671185896985, −8.641931097840436176551190944720, −7.63435173957842756134106323679, −6.30277972219903858027722304272, −5.90319228757082925810056499631, −4.97667142079761360040064708068, −3.79793890353735289619557081300, −3.12895082239690577360446400248, −1.63959696539851249987994299446, −0.34572137987244185856719393328,
0.34572137987244185856719393328, 1.63959696539851249987994299446, 3.12895082239690577360446400248, 3.79793890353735289619557081300, 4.97667142079761360040064708068, 5.90319228757082925810056499631, 6.30277972219903858027722304272, 7.63435173957842756134106323679, 8.641931097840436176551190944720, 9.092818003430239067671185896985
