| L(s) = 1 | + 19.6·3-s + 46.7·5-s + 143.·9-s − 666.·11-s + 650.·13-s + 918.·15-s − 1.18e3·17-s − 1.56e3·19-s + 1.10e3·23-s − 939.·25-s − 1.96e3·27-s + 2.39e3·29-s − 2.04e3·31-s − 1.30e4·33-s + 1.07e3·37-s + 1.27e4·39-s − 1.09e3·41-s − 1.65e4·43-s + 6.68e3·45-s − 8.29e3·47-s − 2.33e4·51-s + 5.51e3·53-s − 3.11e4·55-s − 3.07e4·57-s − 1.42e4·59-s + 1.42e4·61-s + 3.04e4·65-s + ⋯ |
| L(s) = 1 | + 1.26·3-s + 0.836·5-s + 0.588·9-s − 1.65·11-s + 1.06·13-s + 1.05·15-s − 0.996·17-s − 0.994·19-s + 0.433·23-s − 0.300·25-s − 0.518·27-s + 0.529·29-s − 0.382·31-s − 2.09·33-s + 0.129·37-s + 1.34·39-s − 0.102·41-s − 1.36·43-s + 0.492·45-s − 0.547·47-s − 1.25·51-s + 0.269·53-s − 1.38·55-s − 1.25·57-s − 0.532·59-s + 0.489·61-s + 0.892·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 19.6T + 243T^{2} \) |
| 5 | \( 1 - 46.7T + 3.12e3T^{2} \) |
| 11 | \( 1 + 666.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 650.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.18e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.56e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.10e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.04e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.09e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.65e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.29e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 5.51e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.42e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.42e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.97e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.45e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.85e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.06e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 675.T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.25e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.29e4T + 8.58e9T^{2} \) |
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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
−8.866241860365407492421951428571, −8.482048538735100388155159538790, −7.62516687063010358214969054364, −6.49508415947163608567507215202, −5.60207120455315291790207734727, −4.50093232848477509221009277774, −3.29346501538608190757707685264, −2.45718627201263256573661007688, −1.72169686179251972220121097001, 0,
1.72169686179251972220121097001, 2.45718627201263256573661007688, 3.29346501538608190757707685264, 4.50093232848477509221009277774, 5.60207120455315291790207734727, 6.49508415947163608567507215202, 7.62516687063010358214969054364, 8.482048538735100388155159538790, 8.866241860365407492421951428571
