| L(s) = 1 | + 14·3-s + 64·5-s + 49·7-s − 47·9-s − 420·11-s − 860·13-s + 896·15-s + 830·17-s + 490·19-s + 686·21-s − 4.87e3·23-s + 971·25-s − 4.06e3·27-s − 8.75e3·29-s − 5.62e3·31-s − 5.88e3·33-s + 3.13e3·35-s − 1.43e3·37-s − 1.20e4·39-s − 9.25e3·41-s + 1.47e4·43-s − 3.00e3·45-s + 1.01e4·47-s + 2.40e3·49-s + 1.16e4·51-s + 2.30e4·53-s − 2.68e4·55-s + ⋯ |
| L(s) = 1 | + 0.898·3-s + 1.14·5-s + 0.377·7-s − 0.193·9-s − 1.04·11-s − 1.41·13-s + 1.02·15-s + 0.696·17-s + 0.311·19-s + 0.339·21-s − 1.92·23-s + 0.310·25-s − 1.07·27-s − 1.93·29-s − 1.05·31-s − 0.939·33-s + 0.432·35-s − 0.172·37-s − 1.26·39-s − 0.860·41-s + 1.21·43-s − 0.221·45-s + 0.667·47-s + 1/7·49-s + 0.625·51-s + 1.12·53-s − 1.19·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 14 T + p^{5} T^{2} \) |
| 5 | \( 1 - 64 T + p^{5} T^{2} \) |
| 11 | \( 1 + 420 T + p^{5} T^{2} \) |
| 13 | \( 1 + 860 T + p^{5} T^{2} \) |
| 17 | \( 1 - 830 T + p^{5} T^{2} \) |
| 19 | \( 1 - 490 T + p^{5} T^{2} \) |
| 23 | \( 1 + 4872 T + p^{5} T^{2} \) |
| 29 | \( 1 + 8754 T + p^{5} T^{2} \) |
| 31 | \( 1 + 5628 T + p^{5} T^{2} \) |
| 37 | \( 1 + 1434 T + p^{5} T^{2} \) |
| 41 | \( 1 + 9258 T + p^{5} T^{2} \) |
| 43 | \( 1 - 14756 T + p^{5} T^{2} \) |
| 47 | \( 1 - 10108 T + p^{5} T^{2} \) |
| 53 | \( 1 - 23058 T + p^{5} T^{2} \) |
| 59 | \( 1 + 13734 T + p^{5} T^{2} \) |
| 61 | \( 1 - 25352 T + p^{5} T^{2} \) |
| 67 | \( 1 - 19768 T + p^{5} T^{2} \) |
| 71 | \( 1 + 1792 T + p^{5} T^{2} \) |
| 73 | \( 1 - 37914 T + p^{5} T^{2} \) |
| 79 | \( 1 + 95984 T + p^{5} T^{2} \) |
| 83 | \( 1 - 88242 T + p^{5} T^{2} \) |
| 89 | \( 1 - 43762 T + p^{5} T^{2} \) |
| 97 | \( 1 - 65790 T + p^{5} T^{2} \) |
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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
−9.771440832395049466163920814143, −9.081608451124252675555813129672, −7.909643362710055144384162271695, −7.42461513012350669095683574268, −5.75836478299341086903754461088, −5.30698290943596470042090955864, −3.74503925036624730315838257463, −2.42443588552766616876192544828, −1.96938864745208090008454598664, 0,
1.96938864745208090008454598664, 2.42443588552766616876192544828, 3.74503925036624730315838257463, 5.30698290943596470042090955864, 5.75836478299341086903754461088, 7.42461513012350669095683574268, 7.909643362710055144384162271695, 9.081608451124252675555813129672, 9.771440832395049466163920814143
