| L(s) = 1 | − 8·2-s + 48·4-s − 46·5-s − 2·7-s − 256·8-s + 368·10-s − 142·11-s + 16·14-s + 1.28e3·16-s − 2.20e3·20-s + 1.13e3·22-s + 625·25-s − 96·28-s − 1.63e3·29-s + 478·31-s − 6.14e3·32-s + 92·35-s + 1.17e4·40-s − 6.81e3·44-s + 2.40e3·49-s − 5.00e3·50-s + 3.21e3·53-s + 6.53e3·55-s + 512·56-s + 1.30e4·58-s + 1.37e4·59-s − 3.82e3·62-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3·4-s − 1.83·5-s − 0.0408·7-s − 4·8-s + 3.67·10-s − 1.17·11-s + 4/49·14-s + 5·16-s − 5.51·20-s + 2.34·22-s + 25-s − 0.122·28-s − 1.94·29-s + 0.497·31-s − 6·32-s + 0.0751·35-s + 7.35·40-s − 3.52·44-s + 49-s − 2·50-s + 1.14·53-s + 2.15·55-s + 8/49·56-s + 3.89·58-s + 3.94·59-s − 0.994·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1233411849\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1233411849\) |
| \(L(3)\) |
|
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 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 46 T + 1491 T^{2} + 46 p^{4} T^{3} + p^{8} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 2397 T^{2} + 2 p^{4} T^{3} + p^{8} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 142 T + 5523 T^{2} + 142 p^{4} T^{3} + p^{8} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 818 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 478 T - 695037 T^{2} - 478 p^{4} T^{3} + p^{8} T^{4} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 3218 T + 2465043 T^{2} - 3218 p^{4} T^{3} + p^{8} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 6862 T + p^{4} T^{2} )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 8158 T + 38154723 T^{2} - 8158 p^{4} T^{3} + p^{8} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 9118 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 4178 T - 30002637 T^{2} - 4178 p^{4} T^{3} + p^{8} T^{4} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p^{2} T )^{2}( 1 + p^{2} T )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 17282 T + 210138243 T^{2} + 17282 p^{4} T^{3} + p^{8} T^{4} \) |
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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
−11.60326008011844584094451372638, −11.43036703057571306365397663075, −10.72270713245172065713870055473, −10.59824441741333814345343675568, −9.732243151062241894259786776842, −9.628458989034446545917776411308, −8.602729698780058844071265701960, −8.516627439878429229534418035408, −7.915301422674610716783979716484, −7.63657655204489978019863267609, −7.02190195458313042484079414341, −6.85451416857016365094657592322, −5.60875169377138880041991398599, −5.49804612448108627661833033399, −4.04721075624963445720474177748, −3.66299125812256241223754467393, −2.73754134807598678332192923318, −2.17378440101291944553928835667, −1.00458841968366312044476626406, −0.20314215623355876159472177091,
0.20314215623355876159472177091, 1.00458841968366312044476626406, 2.17378440101291944553928835667, 2.73754134807598678332192923318, 3.66299125812256241223754467393, 4.04721075624963445720474177748, 5.49804612448108627661833033399, 5.60875169377138880041991398599, 6.85451416857016365094657592322, 7.02190195458313042484079414341, 7.63657655204489978019863267609, 7.915301422674610716783979716484, 8.516627439878429229534418035408, 8.602729698780058844071265701960, 9.628458989034446545917776411308, 9.732243151062241894259786776842, 10.59824441741333814345343675568, 10.72270713245172065713870055473, 11.43036703057571306365397663075, 11.60326008011844584094451372638
