| L(s) = 1 | + 0.227·2-s − 8.23·3-s − 7.94·4-s − 2.75·5-s − 1.87·6-s + 21.5·7-s − 3.63·8-s + 40.8·9-s − 0.626·10-s + 54.5·11-s + 65.4·12-s − 52.4·13-s + 4.90·14-s + 22.6·15-s + 62.7·16-s + 9.31·18-s − 19.6·19-s + 21.8·20-s − 177.·21-s + 12.4·22-s + 13.9·23-s + 29.9·24-s − 117.·25-s − 11.9·26-s − 114.·27-s − 171.·28-s + 70.0·29-s + ⋯ |
| L(s) = 1 | + 0.0805·2-s − 1.58·3-s − 0.993·4-s − 0.246·5-s − 0.127·6-s + 1.16·7-s − 0.160·8-s + 1.51·9-s − 0.0198·10-s + 1.49·11-s + 1.57·12-s − 1.11·13-s + 0.0936·14-s + 0.390·15-s + 0.980·16-s + 0.121·18-s − 0.237·19-s + 0.244·20-s − 1.84·21-s + 0.120·22-s + 0.126·23-s + 0.254·24-s − 0.939·25-s − 0.0902·26-s − 0.815·27-s − 1.15·28-s + 0.448·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 0.227T + 8T^{2} \) |
| 3 | \( 1 + 8.23T + 27T^{2} \) |
| 5 | \( 1 + 2.75T + 125T^{2} \) |
| 7 | \( 1 - 21.5T + 343T^{2} \) |
| 11 | \( 1 - 54.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 52.4T + 2.19e3T^{2} \) |
| 19 | \( 1 + 19.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.9T + 1.21e4T^{2} \) |
| 29 | \( 1 - 70.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 167.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 198.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 434.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 127.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 207.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 576.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 78.3T + 2.26e5T^{2} \) |
| 67 | \( 1 + 359.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 213.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 29.0T + 3.89e5T^{2} \) |
| 79 | \( 1 - 855.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 13.6T + 5.71e5T^{2} \) |
| 89 | \( 1 + 651.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
| 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
−11.11396694014189204189473513184, −10.07887516338430736146788178061, −9.118838932041731904511178241521, −7.933568468028827441397541546433, −6.78628929826045916517609829367, −5.61429151726650734435822756222, −4.79936941525005952450100214837, −4.05426801494011383856624718381, −1.37992491709147743364544460760, 0,
1.37992491709147743364544460760, 4.05426801494011383856624718381, 4.79936941525005952450100214837, 5.61429151726650734435822756222, 6.78628929826045916517609829367, 7.933568468028827441397541546433, 9.118838932041731904511178241521, 10.07887516338430736146788178061, 11.11396694014189204189473513184
