| L(s) = 1 | − 4.11·2-s + 8.95·4-s + 5·5-s − 20.0·7-s − 3.92·8-s − 20.5·10-s − 1.67·11-s + 11.1·13-s + 82.4·14-s − 55.4·16-s + 8.98·17-s − 50.6·19-s + 44.7·20-s + 6.90·22-s + 214.·23-s + 25·25-s − 45.7·26-s − 179.·28-s − 76.9·29-s − 273.·31-s + 259.·32-s − 37.0·34-s − 100.·35-s − 137.·37-s + 208.·38-s − 19.6·40-s − 53.3·41-s + ⋯ |
| L(s) = 1 | − 1.45·2-s + 1.11·4-s + 0.447·5-s − 1.08·7-s − 0.173·8-s − 0.651·10-s − 0.0460·11-s + 0.237·13-s + 1.57·14-s − 0.866·16-s + 0.128·17-s − 0.611·19-s + 0.500·20-s + 0.0669·22-s + 1.94·23-s + 0.200·25-s − 0.345·26-s − 1.20·28-s − 0.492·29-s − 1.58·31-s + 1.43·32-s − 0.186·34-s − 0.483·35-s − 0.609·37-s + 0.890·38-s − 0.0775·40-s − 0.203·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7148162671\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7148162671\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 2 | \( 1 + 4.11T + 8T^{2} \) |
| 7 | \( 1 + 20.0T + 343T^{2} \) |
| 11 | \( 1 + 1.67T + 1.33e3T^{2} \) |
| 13 | \( 1 - 11.1T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.98T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 214.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 76.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 273.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 53.3T + 6.89e4T^{2} \) |
| 43 | \( 1 - 295.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 194.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 450.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 481.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 675.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 894.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 721.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 915.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 59.6T + 4.93e5T^{2} \) |
| 83 | \( 1 - 742.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.54e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.12e3T + 9.12e5T^{2} \) |
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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
−10.67246831990463520578846425033, −9.623022225398248433112088919455, −9.232502221509482436221642462865, −8.321402668435898269442988762067, −7.14797679243723340600503663531, −6.51957738758768155125729617688, −5.20054403876399269558760989568, −3.50423406384286686147986933497, −2.11012919804982327157643811796, −0.67519877448882438370411732002,
0.67519877448882438370411732002, 2.11012919804982327157643811796, 3.50423406384286686147986933497, 5.20054403876399269558760989568, 6.51957738758768155125729617688, 7.14797679243723340600503663531, 8.321402668435898269442988762067, 9.232502221509482436221642462865, 9.623022225398248433112088919455, 10.67246831990463520578846425033
