| L(s) = 1 | − 32·4-s − 211·7-s − 775·13-s + 1.02e3·16-s + 3.14e3·19-s − 3.12e3·25-s + 6.75e3·28-s − 1.03e4·31-s − 9.88e3·37-s − 3.35e3·43-s + 2.77e4·49-s + 2.48e4·52-s − 1.83e4·61-s − 3.27e4·64-s + 7.34e4·67-s − 7.81e4·73-s − 1.00e5·76-s + 9.70e3·79-s + 1.63e5·91-s − 4.33e4·97-s + 1.00e5·100-s + 7.05e4·103-s + 1.14e5·109-s − 2.16e5·112-s + ⋯ |
| L(s) = 1 | − 4-s − 1.62·7-s − 1.27·13-s + 16-s + 1.99·19-s − 25-s + 1.62·28-s − 1.92·31-s − 1.18·37-s − 0.276·43-s + 1.64·49-s + 1.27·52-s − 0.629·61-s − 64-s + 1.99·67-s − 1.71·73-s − 1.99·76-s + 0.174·79-s + 2.07·91-s − 0.467·97-s + 100-s + 0.655·103-s + 0.922·109-s − 1.62·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| good | 2 | \( 1 + p^{5} T^{2} \) |
| 5 | \( 1 + p^{5} T^{2} \) |
| 7 | \( 1 + 211 T + p^{5} T^{2} \) |
| 11 | \( 1 + p^{5} T^{2} \) |
| 13 | \( 1 + 775 T + p^{5} T^{2} \) |
| 17 | \( 1 + p^{5} T^{2} \) |
| 19 | \( 1 - 3143 T + p^{5} T^{2} \) |
| 23 | \( 1 + p^{5} T^{2} \) |
| 29 | \( 1 + p^{5} T^{2} \) |
| 31 | \( 1 + 10324 T + p^{5} T^{2} \) |
| 37 | \( 1 + 9889 T + p^{5} T^{2} \) |
| 41 | \( 1 + p^{5} T^{2} \) |
| 43 | \( 1 + 3352 T + p^{5} T^{2} \) |
| 47 | \( 1 + p^{5} T^{2} \) |
| 53 | \( 1 + p^{5} T^{2} \) |
| 59 | \( 1 + p^{5} T^{2} \) |
| 61 | \( 1 + 18301 T + p^{5} T^{2} \) |
| 67 | \( 1 - 73475 T + p^{5} T^{2} \) |
| 71 | \( 1 + p^{5} T^{2} \) |
| 73 | \( 1 + 78127 T + p^{5} T^{2} \) |
| 79 | \( 1 - 9707 T + p^{5} T^{2} \) |
| 83 | \( 1 + p^{5} T^{2} \) |
| 89 | \( 1 + p^{5} T^{2} \) |
| 97 | \( 1 + 43339 T + p^{5} T^{2} \) |
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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
−15.79033619680303962906980986372, −14.25342340258505924187739163952, −13.17133707737118963432634873674, −12.15224090194965209031076483406, −9.956856039416352389405811899124, −9.297656412397761840010318031575, −7.34020323923166141964739802817, −5.43717024790662444864372392497, −3.45095995146775490786165261884, 0,
3.45095995146775490786165261884, 5.43717024790662444864372392497, 7.34020323923166141964739802817, 9.297656412397761840010318031575, 9.956856039416352389405811899124, 12.15224090194965209031076483406, 13.17133707737118963432634873674, 14.25342340258505924187739163952, 15.79033619680303962906980986372
