| L(s) = 1 | − 7.21·2-s − 9·3-s + 20.1·4-s − 25·5-s + 64.9·6-s + 147.·7-s + 85.7·8-s + 81·9-s + 180.·10-s + 121·11-s − 181.·12-s − 1.12e3·13-s − 1.06e3·14-s + 225·15-s − 1.26e3·16-s + 1.01e3·17-s − 584.·18-s − 13.7·19-s − 503.·20-s − 1.32e3·21-s − 873.·22-s − 2.11e3·23-s − 771.·24-s + 625·25-s + 8.11e3·26-s − 729·27-s + 2.96e3·28-s + ⋯ |
| L(s) = 1 | − 1.27·2-s − 0.577·3-s + 0.628·4-s − 0.447·5-s + 0.736·6-s + 1.13·7-s + 0.473·8-s + 0.333·9-s + 0.570·10-s + 0.301·11-s − 0.363·12-s − 1.84·13-s − 1.45·14-s + 0.258·15-s − 1.23·16-s + 0.855·17-s − 0.425·18-s − 0.00871·19-s − 0.281·20-s − 0.657·21-s − 0.384·22-s − 0.834·23-s − 0.273·24-s + 0.200·25-s + 2.35·26-s − 0.192·27-s + 0.715·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.6197941677\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6197941677\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 + 7.21T + 32T^{2} \) |
| 7 | \( 1 - 147.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.01e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 13.7T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.11e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.12e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 3.12e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.05e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.39e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.14e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.91e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.73e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.02e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.10e4T + 8.58e9T^{2} \) |
| 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
−11.73904858980933066449765030912, −10.73589125896246282717366652844, −9.941995098414507627497583867431, −8.872857548402799945333864394071, −7.71247386472338710809304547502, −7.24416790679591350418546266865, −5.37706629588292453849612882878, −4.31461018237420543765089994123, −2.00690265717859599537468374719, −0.63533068104857048199077789612,
0.63533068104857048199077789612, 2.00690265717859599537468374719, 4.31461018237420543765089994123, 5.37706629588292453849612882878, 7.24416790679591350418546266865, 7.71247386472338710809304547502, 8.872857548402799945333864394071, 9.941995098414507627497583867431, 10.73589125896246282717366652844, 11.73904858980933066449765030912
