| L(s) = 1 | + 23·3-s + 25·5-s − 49·7-s + 286·9-s − 555·11-s − 241·13-s + 575·15-s − 1.49e3·17-s + 2.03e3·19-s − 1.12e3·21-s + 1.23e3·23-s + 625·25-s + 989·27-s − 5.00e3·29-s − 5.69e3·31-s − 1.27e4·33-s − 1.22e3·35-s − 5.60e3·37-s − 5.54e3·39-s − 2.42e3·41-s − 602·43-s + 7.15e3·45-s + 2.31e4·47-s + 2.40e3·49-s − 3.42e4·51-s − 2.52e4·53-s − 1.38e4·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.447·5-s − 0.377·7-s + 1.17·9-s − 1.38·11-s − 0.395·13-s + 0.659·15-s − 1.25·17-s + 1.29·19-s − 0.557·21-s + 0.484·23-s + 1/5·25-s + 0.261·27-s − 1.10·29-s − 1.06·31-s − 2.04·33-s − 0.169·35-s − 0.672·37-s − 0.583·39-s − 0.225·41-s − 0.0496·43-s + 0.526·45-s + 1.52·47-s + 1/7·49-s − 1.84·51-s − 1.23·53-s − 0.618·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - p^{2} T \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 23 T + p^{5} T^{2} \) |
| 11 | \( 1 + 555 T + p^{5} T^{2} \) |
| 13 | \( 1 + 241 T + p^{5} T^{2} \) |
| 17 | \( 1 + 1491 T + p^{5} T^{2} \) |
| 19 | \( 1 - 2038 T + p^{5} T^{2} \) |
| 23 | \( 1 - 1230 T + p^{5} T^{2} \) |
| 29 | \( 1 + 5001 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5696 T + p^{5} T^{2} \) |
| 37 | \( 1 + 5602 T + p^{5} T^{2} \) |
| 41 | \( 1 + 2424 T + p^{5} T^{2} \) |
| 43 | \( 1 + 14 p T + p^{5} T^{2} \) |
| 47 | \( 1 - 23163 T + p^{5} T^{2} \) |
| 53 | \( 1 + 25296 T + p^{5} T^{2} \) |
| 59 | \( 1 + 5724 T + p^{5} T^{2} \) |
| 61 | \( 1 + 592 p T + p^{5} T^{2} \) |
| 67 | \( 1 + 66104 T + p^{5} T^{2} \) |
| 71 | \( 1 + 16080 T + p^{5} T^{2} \) |
| 73 | \( 1 + 80482 T + p^{5} T^{2} \) |
| 79 | \( 1 - 64147 T + p^{5} T^{2} \) |
| 83 | \( 1 - 106284 T + p^{5} T^{2} \) |
| 89 | \( 1 + 71676 T + p^{5} T^{2} \) |
| 97 | \( 1 - 151025 T + p^{5} T^{2} \) |
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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
−9.271115887691907169241730790413, −8.901140097855542373210648983151, −7.67767962197010789615784543216, −7.24801768182874365032417113879, −5.80188257342631568001332712608, −4.76984118963677800849621250127, −3.42186241102425186488057415551, −2.67838546292522720809543444317, −1.80188253248669264206944961275, 0,
1.80188253248669264206944961275, 2.67838546292522720809543444317, 3.42186241102425186488057415551, 4.76984118963677800849621250127, 5.80188257342631568001332712608, 7.24801768182874365032417113879, 7.67767962197010789615784543216, 8.901140097855542373210648983151, 9.271115887691907169241730790413
