| L(s) = 1 | − 3.02e4·16-s − 8.26e5·31-s − 1.10e7·61-s − 4.78e6·81-s + 7.79e7·121-s + ⋯ |
| L(s) = 1 | − 1.84·16-s − 4.98·31-s − 6.26·61-s − 81-s + 4·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 31640625 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.1304469308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1304469308\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + p^{14} T^{4} \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 + 30233 T^{4} + p^{28} T^{8} \) |
| 7 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
| 17 | $C_2^3$ | \( 1 - 317283271982813182 T^{4} + p^{28} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 - 1653276262 T^{2} + p^{14} T^{4} )^{2} \) |
| 23 | $C_2^3$ | \( 1 - 14019172914771097822 T^{4} + p^{28} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 + 206648 T + p^{7} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + \)\(25\!\cdots\!78\)\( T^{4} + p^{28} T^{8} \) |
| 53 | $C_2^3$ | \( 1 - \)\(21\!\cdots\!22\)\( T^{4} + p^{28} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 61 | $C_2$ | \( ( 1 + 2774518 T + p^{7} T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 38391745250978 T^{2} + p^{14} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + \)\(88\!\cdots\!98\)\( T^{4} + p^{28} T^{8} \) |
| 89 | $C_2$ | \( ( 1 + p^{7} T^{2} )^{4} \) |
| 97 | $C_2^2$ | \( ( 1 + p^{14} T^{4} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.097810909880821389996419157659, −9.033960637752481062348471186936, −8.982836869103587254275226355354, −8.417097929041581166310282557986, −7.953689082831989119586509440130, −7.68402036504414423835060446641, −7.31134739354230428662456831837, −7.05743960641998933821968848359, −7.00826740330135520480648459066, −6.38092020292409928377251746535, −5.97598412342328969346988400086, −5.72070832814890001930576073933, −5.52801794830876309145694778790, −4.87183787180223776551766778678, −4.55804012697039157910345445224, −4.41907289304860359329041185302, −3.84439787105944060654230532038, −3.31795833142219365553115825752, −3.21355206146296599960013213099, −2.58765810004630420099412497444, −1.93208322475447619047354621809, −1.78602438291096058571595919852, −1.49812833711776987354919441314, −0.54308710788122351984340834572, −0.07526664663213171612559683120,
0.07526664663213171612559683120, 0.54308710788122351984340834572, 1.49812833711776987354919441314, 1.78602438291096058571595919852, 1.93208322475447619047354621809, 2.58765810004630420099412497444, 3.21355206146296599960013213099, 3.31795833142219365553115825752, 3.84439787105944060654230532038, 4.41907289304860359329041185302, 4.55804012697039157910345445224, 4.87183787180223776551766778678, 5.52801794830876309145694778790, 5.72070832814890001930576073933, 5.97598412342328969346988400086, 6.38092020292409928377251746535, 7.00826740330135520480648459066, 7.05743960641998933821968848359, 7.31134739354230428662456831837, 7.68402036504414423835060446641, 7.953689082831989119586509440130, 8.417097929041581166310282557986, 8.982836869103587254275226355354, 9.033960637752481062348471186936, 9.097810909880821389996419157659
