| L(s) = 1 | − 4.28·2-s + 9·3-s − 13.6·4-s − 25·5-s − 38.5·6-s + 202.·7-s + 195.·8-s + 81·9-s + 107.·10-s − 121·11-s − 123.·12-s − 636.·13-s − 865.·14-s − 225·15-s − 399.·16-s − 1.20e3·17-s − 346.·18-s + 1.15e3·19-s + 341.·20-s + 1.81e3·21-s + 517.·22-s + 932.·23-s + 1.75e3·24-s + 625·25-s + 2.72e3·26-s + 729·27-s − 2.76e3·28-s + ⋯ |
| L(s) = 1 | − 0.756·2-s + 0.577·3-s − 0.427·4-s − 0.447·5-s − 0.436·6-s + 1.55·7-s + 1.08·8-s + 0.333·9-s + 0.338·10-s − 0.301·11-s − 0.246·12-s − 1.04·13-s − 1.17·14-s − 0.258·15-s − 0.389·16-s − 1.01·17-s − 0.252·18-s + 0.733·19-s + 0.191·20-s + 0.900·21-s + 0.228·22-s + 0.367·23-s + 0.623·24-s + 0.200·25-s + 0.790·26-s + 0.192·27-s − 0.666·28-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
| \(L(3)\) |
\(\approx\) |
\(1.427408671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.427408671\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 4.28T + 32T^{2} \) |
| 7 | \( 1 - 202.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 636.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.20e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 932.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 1.80e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.32e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 6.69e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.35e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.49e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.48e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.80e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.22e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.68e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.25e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.23e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.25e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.28e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.07e4T + 8.58e9T^{2} \) |
| 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
−11.71287688744370479233022467070, −10.80317105688758518689302906627, −9.722868982058642703947935875266, −8.712962198999761316215473137973, −7.967534855716107325440961889082, −7.26833319784118159055536038344, −5.03827111641617236346602690683, −4.29453217913979100516747217342, −2.33644177982430873460179206939, −0.873020326912189159779015848340,
0.873020326912189159779015848340, 2.33644177982430873460179206939, 4.29453217913979100516747217342, 5.03827111641617236346602690683, 7.26833319784118159055536038344, 7.967534855716107325440961889082, 8.712962198999761316215473137973, 9.722868982058642703947935875266, 10.80317105688758518689302906627, 11.71287688744370479233022467070
