| L(s) = 1 | + 9.64·3-s − 86.0·5-s − 149.·9-s + 492.·11-s + 122.·13-s − 830.·15-s + 1.73e3·17-s + 1.99e3·19-s − 2.57e3·23-s + 4.28e3·25-s − 3.79e3·27-s + 3.05e3·29-s − 3.33e3·31-s + 4.74e3·33-s − 1.20e4·37-s + 1.18e3·39-s − 252.·41-s − 1.10e4·43-s + 1.29e4·45-s − 2.25e4·47-s + 1.67e4·51-s − 3.98e4·53-s − 4.23e4·55-s + 1.92e4·57-s − 2.30e4·59-s − 1.12e4·61-s − 1.05e4·65-s + ⋯ |
| L(s) = 1 | + 0.618·3-s − 1.53·5-s − 0.617·9-s + 1.22·11-s + 0.201·13-s − 0.952·15-s + 1.45·17-s + 1.26·19-s − 1.01·23-s + 1.37·25-s − 1.00·27-s + 0.675·29-s − 0.623·31-s + 0.759·33-s − 1.45·37-s + 0.124·39-s − 0.0234·41-s − 0.911·43-s + 0.950·45-s − 1.48·47-s + 0.899·51-s − 1.94·53-s − 1.88·55-s + 0.785·57-s − 0.862·59-s − 0.387·61-s − 0.309·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 9.64T + 243T^{2} \) |
| 5 | \( 1 + 86.0T + 3.12e3T^{2} \) |
| 11 | \( 1 - 492.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 122.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.05e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.20e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 252.T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.25e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.98e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.30e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.12e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.38e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.47e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.26e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.63e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.76e3T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.24e4T + 8.58e9T^{2} \) |
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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
−9.893271243505645024056667714838, −8.961388556164959134962958479516, −8.069459522884151798362522747113, −7.55891808357942598674620236452, −6.32668375647214560504576584195, −4.98483939313241255407482363596, −3.57659258954700408581817439404, −3.34101959981868993245005098127, −1.40534552296451035130666915464, 0,
1.40534552296451035130666915464, 3.34101959981868993245005098127, 3.57659258954700408581817439404, 4.98483939313241255407482363596, 6.32668375647214560504576584195, 7.55891808357942598674620236452, 8.069459522884151798362522747113, 8.961388556164959134962958479516, 9.893271243505645024056667714838
