| L(s) = 1 | − 362·9-s + 200·11-s + 4.48e3·19-s − 1.57e4·29-s + 4.28e3·31-s − 1.48e4·41-s − 1.86e4·49-s − 5.19e4·59-s − 6.11e3·61-s − 7.52e4·71-s − 1.59e5·79-s + 7.19e4·81-s − 1.65e3·89-s − 7.24e4·99-s − 2.87e5·101-s − 2.12e5·109-s − 2.92e5·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2.02e5·169-s + ⋯ |
| L(s) = 1 | − 1.48·9-s + 0.498·11-s + 2.85·19-s − 3.46·29-s + 0.801·31-s − 1.37·41-s − 1.10·49-s − 1.94·59-s − 0.210·61-s − 1.77·71-s − 2.87·79-s + 1.21·81-s − 0.0221·89-s − 0.742·99-s − 2.80·101-s − 1.71·109-s − 1.81·121-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + 3.23e−6·157-s + 2.94e−6·163-s + 2.77e−6·167-s − 0.545·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 362 T^{2} + p^{10} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 18610 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 100 T + p^{5} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 202442 T^{2} + p^{10} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 1879458 T^{2} + p^{10} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 2244 T + p^{5} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 1185810 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7854 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2144 T + p^{5} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 30515770 T^{2} + p^{10} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 7414 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 21442214 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 369731298 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 248174170 T^{2} + p^{10} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 25972 T + p^{5} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 3058 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 755362070 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 37608 T + p^{5} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 3569749522 T^{2} + p^{10} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 79728 T + p^{5} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 7612675530 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 826 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 15761405890 T^{2} + p^{10} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.06476433693374402781690937415, −9.732359392911385592687582414152, −9.211491275119775273157434754493, −9.081652908684144341504772873573, −8.358166833011219750435606864594, −7.922428708031892592764370896946, −7.35675809772460653499090833222, −7.17057755286411460463780794545, −6.28118353401200599702523996805, −5.87206133025554809034082271379, −5.32419860775500449794077016002, −5.18416022551181222557957323432, −4.22229360111555653389876517937, −3.57027638569260245771577678955, −3.09830119560875400915722535667, −2.72629607185546027503657036339, −1.57109475794409627174968063419, −1.33477019010967319602636011112, 0, 0,
1.33477019010967319602636011112, 1.57109475794409627174968063419, 2.72629607185546027503657036339, 3.09830119560875400915722535667, 3.57027638569260245771577678955, 4.22229360111555653389876517937, 5.18416022551181222557957323432, 5.32419860775500449794077016002, 5.87206133025554809034082271379, 6.28118353401200599702523996805, 7.17057755286411460463780794545, 7.35675809772460653499090833222, 7.922428708031892592764370896946, 8.358166833011219750435606864594, 9.081652908684144341504772873573, 9.211491275119775273157434754493, 9.732359392911385592687582414152, 10.06476433693374402781690937415
