| 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
| \(L(4)\) |
\(\approx\) |
\(0.03728763189\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03728763189\) |
| \(L(\frac{9}{2})\) |
|
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
−11.26040570804327509994363585302, −11.13530205436602705172394079036, −10.61706243913378350429667707848, −10.42022227237941806850898957050, −9.354645946453776336684718821393, −9.211773493130211991143738636537, −8.195913969648563519731102312050, −8.009741163420491602537835968433, −7.20379990572559263344193492047, −7.19806969024347981369539628025, −6.09128434337858793643556896668, −5.70389053652259979213004783212, −5.15454827521689041947029312230, −4.74167298248039190523714896813, −3.77238977930306924340965385370, −3.76508115615331126306172884048, −2.29434310319989870698830112531, −1.83199746787973461252024969437, −1.05926535406586398750126083743, −0.05851877363439634932366719804,
0.05851877363439634932366719804, 1.05926535406586398750126083743, 1.83199746787973461252024969437, 2.29434310319989870698830112531, 3.76508115615331126306172884048, 3.77238977930306924340965385370, 4.74167298248039190523714896813, 5.15454827521689041947029312230, 5.70389053652259979213004783212, 6.09128434337858793643556896668, 7.19806969024347981369539628025, 7.20379990572559263344193492047, 8.009741163420491602537835968433, 8.195913969648563519731102312050, 9.211773493130211991143738636537, 9.354645946453776336684718821393, 10.42022227237941806850898957050, 10.61706243913378350429667707848, 11.13530205436602705172394079036, 11.26040570804327509994363585302
