| L(s) = 1 | + 54·3-s − 180·5-s − 1.03e3·7-s + 2.18e3·9-s + 2.84e3·11-s + 340·13-s − 9.72e3·15-s + 9.78e3·17-s + 3.20e4·19-s − 5.57e4·21-s − 1.11e4·23-s − 7.17e4·25-s + 7.87e4·27-s − 3.04e5·29-s + 7.76e4·31-s + 1.53e5·33-s + 1.85e5·35-s − 1.01e6·37-s + 1.83e4·39-s + 7.04e5·41-s − 3.95e5·43-s − 3.93e5·45-s − 1.15e6·47-s + 6.55e5·49-s + 5.28e5·51-s − 1.56e6·53-s − 5.11e5·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.643·5-s − 1.13·7-s + 9-s + 0.643·11-s + 0.0429·13-s − 0.743·15-s + 0.482·17-s + 1.07·19-s − 1.31·21-s − 0.190·23-s − 0.918·25-s + 0.769·27-s − 2.31·29-s + 0.468·31-s + 0.742·33-s + 0.732·35-s − 3.29·37-s + 0.0495·39-s + 1.59·41-s − 0.758·43-s − 0.643·45-s − 1.62·47-s + 0.796·49-s + 0.557·51-s − 1.44·53-s − 0.414·55-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 | $D_{4}$ | \( 1 + 36 p T + 20838 p T^{2} + 36 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 1032 T + 409342 T^{2} + 1032 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 2840 T + 38824982 T^{2} - 2840 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 340 T + 19403694 T^{2} - 340 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 9780 T + 478576006 T^{2} - 9780 p^{7} T^{3} + p^{14} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32040 T + 1749599878 T^{2} - 32040 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 11136 T + 289229518 T^{2} + 11136 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 304212 T + 57634483854 T^{2} + 304212 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 77640 T + 1538948722 p T^{2} - 77640 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 1015820 T + 447827659326 T^{2} + 1015820 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 704100 T + 449624006262 T^{2} - 704100 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 395496 T + 503179893718 T^{2} + 395496 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1157488 T + 1172277002462 T^{2} + 1157488 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 1568580 T + 2748040766334 T^{2} + 1568580 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 40 p^{2} T + 1925517532598 T^{2} - 40 p^{9} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 2603580 T + 5065896220142 T^{2} + 2603580 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 5289768 T + 18885723928102 T^{2} + 5289768 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 5721760 T + 24709125868142 T^{2} + 5721760 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 1190700 T + 17205639157334 T^{2} + 1190700 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 398280 T + 21875776506478 T^{2} - 398280 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 6986616 T + 48072352746118 T^{2} + 6986616 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8166732 T + 74896738657014 T^{2} + 8166732 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10361500 T + 164015274574086 T^{2} + 10361500 p^{7} T^{3} + p^{14} T^{4} \) |
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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
−10.98616858538438537170612415414, −10.34864989668900290612101234244, −9.785893749169103364357865976934, −9.561474851904512813375408342583, −8.935056142133905938490840088820, −8.689713727279348893369282437203, −7.73293713523800053261502582224, −7.60486971419363210483031050716, −7.08427718477604612639810458046, −6.41056389856563389718325532680, −5.80503667832348388339941018401, −5.14260172369309533841908137007, −4.18241482488771285629178448986, −3.80535274720491582520298318201, −3.11244936847674936643152148124, −3.04323680147922211300873316915, −1.62958698373710952628227809618, −1.55427171671921840424117343047, 0, 0,
1.55427171671921840424117343047, 1.62958698373710952628227809618, 3.04323680147922211300873316915, 3.11244936847674936643152148124, 3.80535274720491582520298318201, 4.18241482488771285629178448986, 5.14260172369309533841908137007, 5.80503667832348388339941018401, 6.41056389856563389718325532680, 7.08427718477604612639810458046, 7.60486971419363210483031050716, 7.73293713523800053261502582224, 8.689713727279348893369282437203, 8.935056142133905938490840088820, 9.561474851904512813375408342583, 9.785893749169103364357865976934, 10.34864989668900290612101234244, 10.98616858538438537170612415414
