| L(s) = 1 | − 32·2-s + 640·4-s − 528·5-s + 560·7-s − 1.02e4·8-s + 1.68e4·10-s − 2.16e3·11-s + 1.34e4·13-s − 1.79e4·14-s + 1.43e5·16-s − 2.25e4·17-s + 3.67e4·19-s − 3.37e5·20-s + 6.91e4·22-s − 6.26e4·23-s + 3.04e4·25-s − 4.30e5·26-s + 3.58e5·28-s − 6.84e4·29-s + 2.27e5·31-s − 1.83e6·32-s + 7.21e5·34-s − 2.95e5·35-s + 5.23e5·37-s − 1.17e6·38-s + 5.40e6·40-s − 6.72e4·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 1.88·5-s + 0.617·7-s − 7.07·8-s + 5.34·10-s − 0.489·11-s + 1.69·13-s − 1.74·14-s + 35/4·16-s − 1.11·17-s + 1.22·19-s − 9.44·20-s + 1.38·22-s − 1.07·23-s + 0.390·25-s − 4.80·26-s + 3.08·28-s − 0.520·29-s + 1.37·31-s − 9.89·32-s + 3.15·34-s − 1.16·35-s + 1.69·37-s − 3.47·38-s + 13.3·40-s − 0.152·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{16}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + 528 T + 248294 T^{2} + 15981696 p T^{3} + 1044188259 p^{2} T^{4} + 15981696 p^{8} T^{5} + 248294 p^{14} T^{6} + 528 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 80 p T + 1079188 T^{2} - 27532880 p T^{3} + 673344678406 T^{4} - 27532880 p^{8} T^{5} + 1079188 p^{14} T^{6} - 80 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2160 T + 38790260 T^{2} + 57602735280 T^{3} + 1019380243937334 T^{4} + 57602735280 p^{7} T^{5} + 38790260 p^{14} T^{6} + 2160 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 - 13460 T + 14119498 p T^{2} - 1738650365840 T^{3} + 16613564683301947 T^{4} - 1738650365840 p^{7} T^{5} + 14119498 p^{15} T^{6} - 13460 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 22560 T + 652500158 T^{2} + 256565445120 T^{3} + 29387435907228099 T^{4} + 256565445120 p^{7} T^{5} + 652500158 p^{14} T^{6} + 22560 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 - 36704 T + 3047986756 T^{2} - 91971206881760 T^{3} + 3917212049965000630 T^{4} - 91971206881760 p^{7} T^{5} + 3047986756 p^{14} T^{6} - 36704 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 62640 T + 10480150244 T^{2} + 360495332051760 T^{3} + 1897321084578726474 p T^{4} + 360495332051760 p^{7} T^{5} + 10480150244 p^{14} T^{6} + 62640 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 68400 T + 27591262982 T^{2} - 616022477510400 T^{3} + \)\(39\!\cdots\!95\)\( T^{4} - 616022477510400 p^{7} T^{5} + 27591262982 p^{14} T^{6} + 68400 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 - 227504 T + 73223086396 T^{2} - 5928326820255728 T^{3} + \)\(16\!\cdots\!46\)\( T^{4} - 5928326820255728 p^{7} T^{5} + 73223086396 p^{14} T^{6} - 227504 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 523580 T + 359904908818 T^{2} - 143318985284154800 T^{3} + \)\(13\!\cdots\!83\)\( p T^{4} - 143318985284154800 p^{7} T^{5} + 359904908818 p^{14} T^{6} - 523580 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 67200 T + 520091597828 T^{2} + 37145434869567360 T^{3} + \)\(12\!\cdots\!30\)\( T^{4} + 37145434869567360 p^{7} T^{5} + 520091597828 p^{14} T^{6} + 67200 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 - 562640 T + 737340103492 T^{2} - 276601852750883600 T^{3} + \)\(23\!\cdots\!86\)\( T^{4} - 276601852750883600 p^{7} T^{5} + 737340103492 p^{14} T^{6} - 562640 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 + 515328 T + 1307544720092 T^{2} + 644948319913060608 T^{3} + \)\(92\!\cdots\!38\)\( T^{4} + 644948319913060608 p^{7} T^{5} + 1307544720092 p^{14} T^{6} + 515328 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2498016 T + 4667252414900 T^{2} + 6949193609921122080 T^{3} + \)\(86\!\cdots\!98\)\( T^{4} + 6949193609921122080 p^{7} T^{5} + 4667252414900 p^{14} T^{6} + 2498016 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4155840 T + 15884512231628 T^{2} + 34581097978462526400 T^{3} + \)\(67\!\cdots\!66\)\( T^{4} + 34581097978462526400 p^{7} T^{5} + 15884512231628 p^{14} T^{6} + 4155840 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2130764 T + 7554780642274 T^{2} - 6727373290869154544 T^{3} + \)\(22\!\cdots\!15\)\( T^{4} - 6727373290869154544 p^{7} T^{5} + 7554780642274 p^{14} T^{6} - 2130764 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 1205440 T + 4559355005188 T^{2} + 20411473521454682560 T^{3} + \)\(29\!\cdots\!06\)\( T^{4} + 20411473521454682560 p^{7} T^{5} + 4559355005188 p^{14} T^{6} + 1205440 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12486480 T + 89708735416388 T^{2} + \)\(43\!\cdots\!00\)\( T^{3} + \)\(15\!\cdots\!98\)\( T^{4} + \)\(43\!\cdots\!00\)\( p^{7} T^{5} + 89708735416388 p^{14} T^{6} + 12486480 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4820860 T + 38589071904874 T^{2} + \)\(15\!\cdots\!40\)\( T^{3} + \)\(60\!\cdots\!87\)\( T^{4} + \)\(15\!\cdots\!40\)\( p^{7} T^{5} + 38589071904874 p^{14} T^{6} + 4820860 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 11471680 T + 68478131186836 T^{2} + \)\(28\!\cdots\!60\)\( T^{3} + \)\(12\!\cdots\!86\)\( T^{4} + \)\(28\!\cdots\!60\)\( p^{7} T^{5} + 68478131186836 p^{14} T^{6} + 11471680 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16811232 T + 132213103371692 T^{2} + \)\(81\!\cdots\!40\)\( T^{3} + \)\(45\!\cdots\!06\)\( T^{4} + \)\(81\!\cdots\!40\)\( p^{7} T^{5} + 132213103371692 p^{14} T^{6} + 16811232 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16857600 T + 212149846489550 T^{2} + \)\(19\!\cdots\!00\)\( T^{3} + \)\(14\!\cdots\!99\)\( T^{4} + \)\(19\!\cdots\!00\)\( p^{7} T^{5} + 212149846489550 p^{14} T^{6} + 16857600 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 + 27078040 T + 537918852276700 T^{2} + \)\(68\!\cdots\!40\)\( T^{3} + \)\(72\!\cdots\!26\)\( T^{4} + \)\(68\!\cdots\!40\)\( p^{7} T^{5} + 537918852276700 p^{14} T^{6} + 27078040 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−8.567172455444328542695107329442, −8.100473625524608215363818139229, −8.020161533218158218577496579493, −8.013062901007530611351771709866, −7.889651068700275149714788144804, −7.35201157433738327796747312526, −7.26787633877870209146667512601, −6.83532801976724897536650279082, −6.69216954773100407781264058887, −6.10266754687602312574599926942, −6.00620945808165988110759207462, −5.66382962539088122139955788756, −5.54723371241800316954333999628, −4.57403758359317273179766868475, −4.40746364399037075461460772965, −4.23921448813407975407837364736, −3.91084765377856554516121489971, −3.25821024485899897914048814124, −3.04406663312685026903216246864, −2.64115761557080850562206281362, −2.61286299848058691827103073186, −1.58517687163218254780867683395, −1.52403025331926660872535684985, −1.22684774296893270019825325154, −1.17622267863994156066916097872, 0, 0, 0, 0,
1.17622267863994156066916097872, 1.22684774296893270019825325154, 1.52403025331926660872535684985, 1.58517687163218254780867683395, 2.61286299848058691827103073186, 2.64115761557080850562206281362, 3.04406663312685026903216246864, 3.25821024485899897914048814124, 3.91084765377856554516121489971, 4.23921448813407975407837364736, 4.40746364399037075461460772965, 4.57403758359317273179766868475, 5.54723371241800316954333999628, 5.66382962539088122139955788756, 6.00620945808165988110759207462, 6.10266754687602312574599926942, 6.69216954773100407781264058887, 6.83532801976724897536650279082, 7.26787633877870209146667512601, 7.35201157433738327796747312526, 7.889651068700275149714788144804, 8.013062901007530611351771709866, 8.020161533218158218577496579493, 8.100473625524608215363818139229, 8.567172455444328542695107329442
