| L(s) = 1 | − 9·2-s − 14·3-s − 197·4-s − 222·5-s + 126·6-s − 1.24e3·7-s + 1.21e3·8-s − 6.72e3·9-s + 1.99e3·10-s − 8.71e3·11-s + 2.75e3·12-s − 4.48e3·13-s + 1.12e4·14-s + 3.10e3·15-s + 2.97e4·16-s − 4.44e3·17-s + 6.05e4·18-s + 2.74e4·19-s + 4.37e4·20-s + 1.74e4·21-s + 7.84e4·22-s − 3.05e4·23-s − 1.70e4·24-s − 1.43e5·25-s + 4.03e4·26-s + 8.02e4·27-s + 2.45e5·28-s + ⋯ |
| L(s) = 1 | − 0.795·2-s − 0.299·3-s − 1.53·4-s − 0.794·5-s + 0.238·6-s − 1.37·7-s + 0.838·8-s − 3.07·9-s + 0.631·10-s − 1.97·11-s + 0.460·12-s − 0.565·13-s + 1.09·14-s + 0.237·15-s + 1.81·16-s − 0.219·17-s + 2.44·18-s + 0.917·19-s + 1.22·20-s + 0.411·21-s + 1.57·22-s − 0.523·23-s − 0.251·24-s − 1.83·25-s + 0.449·26-s + 0.785·27-s + 2.11·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 130321 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 19 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + 9 T + 139 p T^{2} + 765 p^{2} T^{3} + 5205 p^{3} T^{4} + 765 p^{9} T^{5} + 139 p^{15} T^{6} + 9 p^{21} T^{7} + p^{28} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + 14 T + 769 p^{2} T^{2} + 4102 p^{3} T^{3} + 770008 p^{3} T^{4} + 4102 p^{10} T^{5} + 769 p^{16} T^{6} + 14 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 222 T + 38569 p T^{2} + 2210238 p^{2} T^{3} + 151265856 p^{3} T^{4} + 2210238 p^{9} T^{5} + 38569 p^{15} T^{6} + 222 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 178 p T + 2457124 T^{2} + 302848984 p T^{3} + 2721611058013 T^{4} + 302848984 p^{8} T^{5} + 2457124 p^{14} T^{6} + 178 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 8718 T + 7577131 p T^{2} + 397129760262 T^{3} + 2253310600559136 T^{4} + 397129760262 p^{7} T^{5} + 7577131 p^{15} T^{6} + 8718 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 4480 T + 10442431 p T^{2} + 1216594982680 T^{3} + 8905550799721468 T^{4} + 1216594982680 p^{7} T^{5} + 10442431 p^{15} T^{6} + 4480 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 4440 T + 779869562 T^{2} - 4771177186320 T^{3} + 272572121869323963 T^{4} - 4771177186320 p^{7} T^{5} + 779869562 p^{14} T^{6} + 4440 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 30528 T + 6766373099 T^{2} + 281312350347456 T^{3} + 30869060512593572088 T^{4} + 281312350347456 p^{7} T^{5} + 6766373099 p^{14} T^{6} + 30528 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 254244 T + 50526434375 T^{2} + 4158825389649144 T^{3} + \)\(56\!\cdots\!48\)\( T^{4} + 4158825389649144 p^{7} T^{5} + 50526434375 p^{14} T^{6} + 254244 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 303460 T + 2479413988 p T^{2} + 13630700228047828 T^{3} + \)\(27\!\cdots\!06\)\( T^{4} + 13630700228047828 p^{7} T^{5} + 2479413988 p^{15} T^{6} + 303460 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 270460 T + 190182623716 T^{2} + 33517908738047860 T^{3} + \)\(19\!\cdots\!78\)\( T^{4} + 33517908738047860 p^{7} T^{5} + 190182623716 p^{14} T^{6} + 270460 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 828564 T + 615809837432 T^{2} + 263636106124383612 T^{3} + \)\(13\!\cdots\!62\)\( T^{4} + 263636106124383612 p^{7} T^{5} + 615809837432 p^{14} T^{6} + 828564 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 37454 T + 745997453341 T^{2} + 58604976088183150 T^{3} + \)\(25\!\cdots\!72\)\( T^{4} + 58604976088183150 p^{7} T^{5} + 745997453341 p^{14} T^{6} - 37454 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 335670 T + 80999396117 T^{2} + 82836172922388930 T^{3} + \)\(13\!\cdots\!48\)\( T^{4} + 82836172922388930 p^{7} T^{5} + 80999396117 p^{14} T^{6} - 335670 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 76728 T + 3451678167515 T^{2} - 456735752197732368 T^{3} + \)\(55\!\cdots\!72\)\( T^{4} - 456735752197732368 p^{7} T^{5} + 3451678167515 p^{14} T^{6} - 76728 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 3191334 T + 9572074588361 T^{2} + 18383692890303807138 T^{3} + \)\(32\!\cdots\!96\)\( T^{4} + 18383692890303807138 p^{7} T^{5} + 9572074588361 p^{14} T^{6} + 3191334 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 346550 T + 10546139924665 T^{2} - 3501286532199067106 T^{3} + \)\(47\!\cdots\!20\)\( T^{4} - 3501286532199067106 p^{7} T^{5} + 10546139924665 p^{14} T^{6} - 346550 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 270322 T + 9026732328853 T^{2} - 10148100690291020534 T^{3} + \)\(32\!\cdots\!48\)\( T^{4} - 10148100690291020534 p^{7} T^{5} + 9026732328853 p^{14} T^{6} + 270322 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 2066124 T + 18123756199196 T^{2} + 31467662292373515468 T^{3} + \)\(15\!\cdots\!42\)\( T^{4} + 31467662292373515468 p^{7} T^{5} + 18123756199196 p^{14} T^{6} + 2066124 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 416044 T + 27420631550362 T^{2} + 7627031467546596208 T^{3} + \)\(42\!\cdots\!63\)\( T^{4} + 7627031467546596208 p^{7} T^{5} + 27420631550362 p^{14} T^{6} + 416044 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 16025864 T + 165936409446616 T^{2} - \)\(11\!\cdots\!40\)\( T^{3} + \)\(57\!\cdots\!46\)\( T^{4} - \)\(11\!\cdots\!40\)\( p^{7} T^{5} + 165936409446616 p^{14} T^{6} - 16025864 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 8524128 T + 119138452816904 T^{2} - \)\(65\!\cdots\!20\)\( T^{3} + \)\(49\!\cdots\!66\)\( T^{4} - \)\(65\!\cdots\!20\)\( p^{7} T^{5} + 119138452816904 p^{14} T^{6} - 8524128 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 2899092 T + 139779480449732 T^{2} - \)\(34\!\cdots\!16\)\( T^{3} + \)\(86\!\cdots\!54\)\( T^{4} - \)\(34\!\cdots\!16\)\( p^{7} T^{5} + 139779480449732 p^{14} T^{6} - 2899092 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 4766908 T + 110311755135064 T^{2} + \)\(75\!\cdots\!60\)\( T^{3} + \)\(86\!\cdots\!82\)\( T^{4} + \)\(75\!\cdots\!60\)\( p^{7} T^{5} + 110311755135064 p^{14} T^{6} + 4766908 p^{21} T^{7} + p^{28} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.36153272833391726159767536940, −12.51382454827112294637115858466, −12.27853989033566855699439557329, −12.20255731670735797416615345243, −11.52995355556944873493640779296, −11.24194390156311109029488627525, −10.95082667044089614081906507840, −10.39015559826478387606945877457, −9.998942305959373118290961247891, −9.406087985482269555575881065000, −9.305010816885172165202191815402, −9.101066792903639275410024048666, −8.572233735888957194970151729705, −7.947937172009549844828116499513, −7.891752060594996440077031514104, −7.68700989944601172530915363324, −6.61730972335786549023836858023, −6.05952201512752652457829302237, −5.45443163196034056463253453140, −5.42800639221185321587342495500, −4.99695903503744192178456194094, −3.71037952877483972611751057178, −3.50683639537017352707305423611, −2.99292688189522424754034179389, −2.19378523249737211995813802187, 0, 0, 0, 0,
2.19378523249737211995813802187, 2.99292688189522424754034179389, 3.50683639537017352707305423611, 3.71037952877483972611751057178, 4.99695903503744192178456194094, 5.42800639221185321587342495500, 5.45443163196034056463253453140, 6.05952201512752652457829302237, 6.61730972335786549023836858023, 7.68700989944601172530915363324, 7.891752060594996440077031514104, 7.947937172009549844828116499513, 8.572233735888957194970151729705, 9.101066792903639275410024048666, 9.305010816885172165202191815402, 9.406087985482269555575881065000, 9.998942305959373118290961247891, 10.39015559826478387606945877457, 10.95082667044089614081906507840, 11.24194390156311109029488627525, 11.52995355556944873493640779296, 12.20255731670735797416615345243, 12.27853989033566855699439557329, 12.51382454827112294637115858466, 13.36153272833391726159767536940
