| L(s) = 1 | − 54·3-s + 28·5-s − 936·7-s + 2.18e3·9-s − 3.38e3·11-s + 1.70e4·13-s − 1.51e3·15-s − 1.86e4·17-s + 4.48e4·19-s + 5.05e4·21-s − 4.34e4·23-s − 3.12e4·25-s − 7.87e4·27-s + 1.94e4·29-s − 3.32e5·31-s + 1.82e5·33-s − 2.62e4·35-s + 9.07e3·37-s − 9.22e5·39-s + 4.91e5·41-s + 5.10e5·43-s + 6.12e4·45-s − 1.78e6·47-s − 3.12e5·49-s + 1.00e6·51-s + 1.39e6·53-s − 9.47e4·55-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.100·5-s − 1.03·7-s + 9-s − 0.766·11-s + 2.15·13-s − 0.115·15-s − 0.921·17-s + 1.50·19-s + 1.19·21-s − 0.745·23-s − 0.399·25-s − 0.769·27-s + 0.148·29-s − 2.00·31-s + 0.885·33-s − 0.103·35-s + 0.0294·37-s − 2.48·39-s + 1.11·41-s + 0.980·43-s + 0.100·45-s − 2.50·47-s − 0.379·49-s + 1.06·51-s + 1.28·53-s − 0.0767·55-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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $D_{4}$ | \( 1 - 28 T + 6406 p T^{2} - 28 p^{7} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 936 T + 1188734 T^{2} + 936 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 3384 T + 12585622 T^{2} + 3384 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 17076 T + 174008942 T^{2} - 17076 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 18668 T + 763976006 T^{2} + 18668 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 44856 T + 2237619206 T^{2} - 44856 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 43488 T + 1865158606 T^{2} + 43488 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 19484 T + 563275118 T^{2} - 19484 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 332856 T + 60250867406 T^{2} + 332856 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 9076 T + 83660952894 T^{2} - 9076 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 491780 T + 442441281206 T^{2} - 491780 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 510984 T + 247939713878 T^{2} - 510984 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1781424 T + 1786374389470 T^{2} + 1781424 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 1395692 T + 2682288635486 T^{2} - 1395692 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 1534104 T + 5559576794038 T^{2} - 1534104 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1592188 T + 6774132506478 T^{2} + 1592188 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1169496 T - 1041831063514 T^{2} + 1169496 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5716800 T + 23877712415086 T^{2} + 5716800 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 1180884 T + 15089768314454 T^{2} - 1180884 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 6538104 T + 48301289191022 T^{2} + 6538104 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 16805160 T + 122017660191238 T^{2} + 16805160 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 6118924 T + 91061417808758 T^{2} + 6118924 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 23720868 T + 287915376622982 T^{2} - 23720868 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−11.14646593487165981350160035829, −10.52928043253335344299832105621, −10.09261273768634625519763126852, −9.708766750903103146061290951339, −8.853766085694663127367934332651, −8.804858774849473182387272941986, −7.66224334059995851659278257446, −7.49764386044348888832843548818, −6.60924320832307520374685237766, −6.29960472513332279358358937564, −5.61863932145410744980402787626, −5.56358033686033007544496644508, −4.56207958852901442730845078202, −3.89026168484771206369926630970, −3.42324934429685903597654080596, −2.64388946919992546141170401693, −1.58872695378975618013156499611, −1.13287819142144672799856266170, 0, 0,
1.13287819142144672799856266170, 1.58872695378975618013156499611, 2.64388946919992546141170401693, 3.42324934429685903597654080596, 3.89026168484771206369926630970, 4.56207958852901442730845078202, 5.56358033686033007544496644508, 5.61863932145410744980402787626, 6.29960472513332279358358937564, 6.60924320832307520374685237766, 7.49764386044348888832843548818, 7.66224334059995851659278257446, 8.804858774849473182387272941986, 8.853766085694663127367934332651, 9.708766750903103146061290951339, 10.09261273768634625519763126852, 10.52928043253335344299832105621, 11.14646593487165981350160035829
