L(s) = 1 | + 54·3-s − 196·5-s − 504·7-s + 2.18e3·9-s + 1.65e3·11-s − 6.15e3·13-s − 1.05e4·15-s + 1.71e4·17-s − 504·19-s − 2.72e4·21-s − 5.15e4·23-s − 2.14e4·25-s + 7.87e4·27-s + 1.99e5·29-s − 2.57e5·31-s + 8.94e4·33-s + 9.87e4·35-s + 4.68e5·37-s − 3.32e5·39-s − 1.06e5·41-s + 1.61e6·43-s − 4.28e5·45-s − 6.46e5·47-s − 5.02e5·49-s + 9.23e5·51-s − 1.46e6·53-s − 3.24e5·55-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.701·5-s − 0.555·7-s + 9-s + 0.375·11-s − 0.777·13-s − 0.809·15-s + 0.844·17-s − 0.0168·19-s − 0.641·21-s − 0.883·23-s − 0.274·25-s + 0.769·27-s + 1.52·29-s − 1.55·31-s + 0.433·33-s + 0.389·35-s + 1.52·37-s − 0.897·39-s − 0.242·41-s + 3.10·43-s − 0.701·45-s − 0.908·47-s − 0.610·49-s + 0.975·51-s − 1.35·53-s − 0.263·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(3.648208964\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.648208964\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
good | 5 | $D_{4}$ | \( 1 + 196 T + 11974 p T^{2} + 196 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 72 p T + 756734 T^{2} + 72 p^{8} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 1656 T + 35844502 T^{2} - 1656 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 6156 T + 134547182 T^{2} + 6156 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 17108 T + 486445766 T^{2} - 17108 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 504 T - 230552314 T^{2} + 504 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 51552 T + 7470237646 T^{2} + 51552 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 199804 T + 32156236718 T^{2} - 199804 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 257256 T + 69638832206 T^{2} + 257256 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 468724 T + 243553103934 T^{2} - 468724 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 106940 T + 285959228726 T^{2} + 106940 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 1617336 T + 1147122174038 T^{2} - 1617336 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 646416 T + 220226541790 T^{2} + 646416 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1469492 T + 1664819509406 T^{2} + 1469492 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 4541544 T + 9906553362358 T^{2} - 4541544 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 7892 p T + 286057908078 T^{2} - 7892 p^{8} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4775256 T + 15866190670886 T^{2} - 4775256 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1094400 T + 3637256504686 T^{2} - 1094400 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 5731884 T + 372597068198 p T^{2} + 5731884 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 10402776 T + 57467964116462 T^{2} - 10402776 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2212200 T + 33756154600198 T^{2} - 2212200 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3604364 T + 80061377244278 T^{2} + 3604364 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 7156188 T + 162174325590662 T^{2} + 7156188 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.31985865091499847401543911327, −11.24083666351970386582887941960, −10.20565704701933492852205237785, −9.975360451634325881842137912065, −9.462209534053238644969582028524, −9.110164568275481413817547793101, −8.281718195853893395880278052325, −8.096394304144746429303253030070, −7.43625692238067919758258477707, −7.17563423912580281261305530235, −6.36061867721738699988513067191, −5.86449329472675767302835346336, −4.97161471212477710085636156966, −4.36943244910672222331985892154, −3.68994063153019109085578076875, −3.44572562012960974054556228570, −2.51067641928418375318586327457, −2.18505674684068188566291474748, −1.09381648684703272650126100422, −0.49639429830121355748545167606,
0.49639429830121355748545167606, 1.09381648684703272650126100422, 2.18505674684068188566291474748, 2.51067641928418375318586327457, 3.44572562012960974054556228570, 3.68994063153019109085578076875, 4.36943244910672222331985892154, 4.97161471212477710085636156966, 5.86449329472675767302835346336, 6.36061867721738699988513067191, 7.17563423912580281261305530235, 7.43625692238067919758258477707, 8.096394304144746429303253030070, 8.281718195853893395880278052325, 9.110164568275481413817547793101, 9.462209534053238644969582028524, 9.975360451634325881842137912065, 10.20565704701933492852205237785, 11.24083666351970386582887941960, 11.31985865091499847401543911327