| 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
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
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 | $D_{4}$ | \( 1 + 196 T + 11974 p T^{2} + 196 p^{7} T^{3} + p^{14} T^{4} \) |
| 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 | $D_{4}$ | \( 1 + 7156188 T + 162174325590662 T^{2} + 7156188 p^{7} T^{3} + p^{14} T^{4} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.14606143098310637659171420403, −10.54420036166145543052335423437, −9.965893306344905434399360743511, −9.923973174986335929024765842023, −8.971752380786676369985919011135, −8.327446015123724384960436002020, −7.83344999033741279427543343157, −7.52120824862848834478978231187, −6.79288338035223477833794220078, −6.33155354500302120823043225788, −5.71826384866900036743757011922, −5.02901358371498125270825719079, −4.58559776361689209969241659913, −4.33595144610211452533701805038, −3.04566381049806049076275576421, −2.86684930199100401529961763994, −1.43215018083045356291091973637, −1.23498449453113110289646965218, 0, 0,
1.23498449453113110289646965218, 1.43215018083045356291091973637, 2.86684930199100401529961763994, 3.04566381049806049076275576421, 4.33595144610211452533701805038, 4.58559776361689209969241659913, 5.02901358371498125270825719079, 5.71826384866900036743757011922, 6.33155354500302120823043225788, 6.79288338035223477833794220078, 7.52120824862848834478978231187, 7.83344999033741279427543343157, 8.327446015123724384960436002020, 8.971752380786676369985919011135, 9.923973174986335929024765842023, 9.965893306344905434399360743511, 10.54420036166145543052335423437, 11.14606143098310637659171420403
