L(s) = 1 | + 1.48·2-s + 9·3-s − 29.7·4-s + 13.3·6-s + 7.94·7-s − 91.9·8-s + 81·9-s − 121·11-s − 268.·12-s − 541.·13-s + 11.8·14-s + 816.·16-s + 1.07e3·17-s + 120.·18-s + 809.·19-s + 71.4·21-s − 180.·22-s − 614.·23-s − 827.·24-s − 805.·26-s + 729·27-s − 236.·28-s − 552.·29-s + 9.76e3·31-s + 4.15e3·32-s − 1.08e3·33-s + 1.59e3·34-s + ⋯ |
L(s) = 1 | + 0.263·2-s + 0.577·3-s − 0.930·4-s + 0.151·6-s + 0.0612·7-s − 0.507·8-s + 0.333·9-s − 0.301·11-s − 0.537·12-s − 0.888·13-s + 0.0161·14-s + 0.797·16-s + 0.898·17-s + 0.0876·18-s + 0.514·19-s + 0.0353·21-s − 0.0793·22-s − 0.242·23-s − 0.293·24-s − 0.233·26-s + 0.192·27-s − 0.0570·28-s − 0.121·29-s + 1.82·31-s + 0.717·32-s − 0.174·33-s + 0.236·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
good | 2 | \( 1 - 1.48T + 32T^{2} \) |
| 7 | \( 1 - 7.94T + 1.68e4T^{2} \) |
| 13 | \( 1 + 541.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 809.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 614.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 552.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 9.76e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 839.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.12e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.56e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.03e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.16e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.64e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.27e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.40e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 2.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 7.91e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.32e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.13e5T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.57e3T + 8.58e9T^{2} \) |
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
−9.106945689662244708364545606649, −8.141086191232257951608904807460, −7.63280484306165743646850212502, −6.37619762155957936436907458956, −5.24647250311313720643563922437, −4.61195585779719093699416138108, −3.51570275816824800943090721176, −2.69524565772462769361503746119, −1.24579311250788875004589322948, 0,
1.24579311250788875004589322948, 2.69524565772462769361503746119, 3.51570275816824800943090721176, 4.61195585779719093699416138108, 5.24647250311313720643563922437, 6.37619762155957936436907458956, 7.63280484306165743646850212502, 8.141086191232257951608904807460, 9.106945689662244708364545606649