| L(s) = 1 | + 10·5-s + 42·11-s − 76·13-s + 78·17-s − 162·19-s + 48·23-s − 219·25-s + 88·29-s − 16·31-s − 406·37-s + 123·41-s − 236·43-s − 528·47-s − 615·49-s + 92·53-s + 420·55-s − 712·59-s − 394·61-s − 760·65-s + 446·67-s − 804·71-s + 298·73-s + 936·79-s − 796·83-s + 780·85-s + 314·89-s − 1.62e3·95-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 1.15·11-s − 1.62·13-s + 1.11·17-s − 1.95·19-s + 0.435·23-s − 1.75·25-s + 0.563·29-s − 0.0926·31-s − 1.80·37-s + 0.468·41-s − 0.836·43-s − 1.63·47-s − 1.79·49-s + 0.238·53-s + 1.02·55-s − 1.57·59-s − 0.826·61-s − 1.45·65-s + 0.813·67-s − 1.34·71-s + 0.477·73-s + 1.33·79-s − 1.05·83-s + 0.995·85-s + 0.373·89-s − 1.74·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 41^{3}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 41 | $C_1$ | \( ( 1 - p T )^{3} \) |
| good | 5 | $S_4\times C_2$ | \( 1 - 2 p T + 319 T^{2} - 2548 T^{3} + 319 p^{3} T^{4} - 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 615 T^{2} - 2224 T^{3} + 615 p^{3} T^{4} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 - 42 T + 4023 T^{2} - 111712 T^{3} + 4023 p^{3} T^{4} - 42 p^{6} T^{5} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 + 76 T + 635 p T^{2} + 343672 T^{3} + 635 p^{4} T^{4} + 76 p^{6} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 - 78 T + 13551 T^{2} - 633700 T^{3} + 13551 p^{3} T^{4} - 78 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 162 T + 23391 T^{2} + 1884688 T^{3} + 23391 p^{3} T^{4} + 162 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 48 T + 32733 T^{2} - 1007904 T^{3} + 32733 p^{3} T^{4} - 48 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 - 88 T + 44799 T^{2} - 2058352 T^{3} + 44799 p^{3} T^{4} - 88 p^{6} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 16 T + 78181 T^{2} + 1083488 T^{3} + 78181 p^{3} T^{4} + 16 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 + 406 T + 171583 T^{2} + 37174604 T^{3} + 171583 p^{3} T^{4} + 406 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 + 236 T + 97865 T^{2} + 15882440 T^{3} + 97865 p^{3} T^{4} + 236 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 + 528 T + 334431 T^{2} + 107660976 T^{3} + 334431 p^{3} T^{4} + 528 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 - 92 T + 319191 T^{2} - 18806936 T^{3} + 319191 p^{3} T^{4} - 92 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 + 712 T + 754513 T^{2} + 299112304 T^{3} + 754513 p^{3} T^{4} + 712 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 + 394 T + 592907 T^{2} + 182836732 T^{3} + 592907 p^{3} T^{4} + 394 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 446 T + 904143 T^{2} - 267507424 T^{3} + 904143 p^{3} T^{4} - 446 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 804 T + 803319 T^{2} + 341218560 T^{3} + 803319 p^{3} T^{4} + 804 p^{6} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 - 298 T + 760395 T^{2} - 93211700 T^{3} + 760395 p^{3} T^{4} - 298 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 - 936 T + 1593951 T^{2} - 916096176 T^{3} + 1593951 p^{3} T^{4} - 936 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 + 796 T + 1497009 T^{2} + 871822120 T^{3} + 1497009 p^{3} T^{4} + 796 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 - 314 T + 1224583 T^{2} - 688444364 T^{3} + 1224583 p^{3} T^{4} - 314 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 302 T + 408815 T^{2} - 851519164 T^{3} + 408815 p^{3} T^{4} + 302 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.553540707010780296501340783173, −8.034762270207847667752650995070, −7.993494778844343239856734844734, −7.81705880838411024370702684937, −7.25726449663147035225033172130, −7.19896639472836840639279666575, −6.72195697236830977734144135861, −6.45240640514954451992363053667, −6.28619339718885511587959347731, −6.25151198204738693010381562413, −5.51816503647282330592453193901, −5.51508086249409465127228461844, −5.20877094150481314961707746348, −4.66795335349293934553446065280, −4.54626428615890486674779586435, −4.43174492981586899446177394624, −3.60738956968462574954487630063, −3.54171782786046200571359244445, −3.46077720285572075625779818419, −2.69594594015684398232372778124, −2.35294307478122024239336504796, −2.26468430223264600324601602687, −1.59441407332108142714759742449, −1.45100710657762457199556474235, −1.21885267151754866375672747125, 0, 0, 0,
1.21885267151754866375672747125, 1.45100710657762457199556474235, 1.59441407332108142714759742449, 2.26468430223264600324601602687, 2.35294307478122024239336504796, 2.69594594015684398232372778124, 3.46077720285572075625779818419, 3.54171782786046200571359244445, 3.60738956968462574954487630063, 4.43174492981586899446177394624, 4.54626428615890486674779586435, 4.66795335349293934553446065280, 5.20877094150481314961707746348, 5.51508086249409465127228461844, 5.51816503647282330592453193901, 6.25151198204738693010381562413, 6.28619339718885511587959347731, 6.45240640514954451992363053667, 6.72195697236830977734144135861, 7.19896639472836840639279666575, 7.25726449663147035225033172130, 7.81705880838411024370702684937, 7.993494778844343239856734844734, 8.034762270207847667752650995070, 8.553540707010780296501340783173
