| L(s) = 1 | − 3.46·2-s + 3.99·4-s − 13.8·5-s − 22.5·7-s + 13.8·8-s + 47.9·10-s + 22.5·11-s + 77.9·14-s − 80·16-s + 27·17-s − 88.3·19-s − 55.4·20-s − 77.9·22-s + 57·23-s + 66.9·25-s − 90.0·28-s + 69·29-s − 72.7·31-s + 166.·32-s − 93.5·34-s + 311.·35-s − 39.8·37-s + 306·38-s − 192.·40-s − 393.·41-s + 85·43-s + 90.0·44-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.499·4-s − 1.23·5-s − 1.21·7-s + 0.612·8-s + 1.51·10-s + 0.617·11-s + 1.48·14-s − 1.25·16-s + 0.385·17-s − 1.06·19-s − 0.619·20-s − 0.755·22-s + 0.516·23-s + 0.535·25-s − 0.607·28-s + 0.441·29-s − 0.421·31-s + 0.918·32-s − 0.471·34-s + 1.50·35-s − 0.177·37-s + 1.30·38-s − 0.758·40-s − 1.49·41-s + 0.301·43-s + 0.308·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1829400896\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1829400896\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 3.46T + 8T^{2} \) |
| 5 | \( 1 + 13.8T + 125T^{2} \) |
| 7 | \( 1 + 22.5T + 343T^{2} \) |
| 11 | \( 1 - 22.5T + 1.33e3T^{2} \) |
| 17 | \( 1 - 27T + 4.91e3T^{2} \) |
| 19 | \( 1 + 88.3T + 6.85e3T^{2} \) |
| 23 | \( 1 - 57T + 1.21e4T^{2} \) |
| 29 | \( 1 - 69T + 2.43e4T^{2} \) |
| 31 | \( 1 + 72.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + 39.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 393.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 85T + 7.95e4T^{2} \) |
| 47 | \( 1 + 342.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 426T + 1.48e5T^{2} \) |
| 59 | \( 1 + 19.0T + 2.05e5T^{2} \) |
| 61 | \( 1 + 17T + 2.26e5T^{2} \) |
| 67 | \( 1 + 164.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 583.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.00e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.24e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 426.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 306.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.23e3T + 9.12e5T^{2} \) |
| 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
−8.962254975805156522841103640230, −8.485118458944135114586963788419, −7.62468029585644588431638466048, −6.94479167643991053400292848981, −6.23500934074067587912894670163, −4.75200509128778970915759694154, −3.89151148736981725006837414788, −3.04593588660560850927418347817, −1.51608997029439906082986117727, −0.25975404239308756247164309666,
0.25975404239308756247164309666, 1.51608997029439906082986117727, 3.04593588660560850927418347817, 3.89151148736981725006837414788, 4.75200509128778970915759694154, 6.23500934074067587912894670163, 6.94479167643991053400292848981, 7.62468029585644588431638466048, 8.485118458944135114586963788419, 8.962254975805156522841103640230
