| L(s) = 1 | + 32·2-s + 108·3-s + 640·4-s − 332·5-s + 3.45e3·6-s − 1.37e3·7-s + 1.02e4·8-s + 7.29e3·9-s − 1.06e4·10-s + 5.32e3·11-s + 6.91e4·12-s − 2.56e3·13-s − 4.39e4·14-s − 3.58e4·15-s + 1.43e5·16-s − 2.99e4·17-s + 2.33e5·18-s − 5.34e4·19-s − 2.12e5·20-s − 1.48e5·21-s + 1.70e5·22-s − 8.02e4·23-s + 1.10e6·24-s − 9.38e4·25-s − 8.21e4·26-s + 3.93e5·27-s − 8.78e5·28-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 2.30·3-s + 5·4-s − 1.18·5-s + 6.53·6-s − 1.51·7-s + 7.07·8-s + 10/3·9-s − 3.35·10-s + 1.20·11-s + 11.5·12-s − 0.323·13-s − 4.27·14-s − 2.74·15-s + 35/4·16-s − 1.48·17-s + 9.42·18-s − 1.78·19-s − 5.93·20-s − 3.49·21-s + 3.41·22-s − 1.37·23-s + 16.3·24-s − 1.20·25-s − 0.916·26-s + 3.84·27-s − 7.55·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 332 T + 204033 T^{2} + 9952446 p T^{3} + 807451048 p^{2} T^{4} + 9952446 p^{8} T^{5} + 204033 p^{14} T^{6} + 332 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2566 T + 147539625 T^{2} - 32898488062 T^{3} + 10181486967169636 T^{4} - 32898488062 p^{7} T^{5} + 147539625 p^{14} T^{6} + 2566 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 29996 T + 671899580 T^{2} + 16373556277180 T^{3} + 385690807635191414 T^{4} + 16373556277180 p^{7} T^{5} + 671899580 p^{14} T^{6} + 29996 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 53442 T + 3022469959 T^{2} + 110902713953832 T^{3} + 3971082228676036656 T^{4} + 110902713953832 p^{7} T^{5} + 3022469959 p^{14} T^{6} + 53442 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 80238 T + 7248774672 T^{2} + 304390396709574 T^{3} + 25561462166222223614 T^{4} + 304390396709574 p^{7} T^{5} + 7248774672 p^{14} T^{6} + 80238 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 248558 T + 64898830741 T^{2} + 8685843400723090 T^{3} + \)\(14\!\cdots\!52\)\( T^{4} + 8685843400723090 p^{7} T^{5} + 64898830741 p^{14} T^{6} + 248558 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 464196 T + 142114731340 T^{2} + 30435407127310556 T^{3} + \)\(54\!\cdots\!62\)\( T^{4} + 30435407127310556 p^{7} T^{5} + 142114731340 p^{14} T^{6} + 464196 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 389046 T + 292914435841 T^{2} + 93928718675848490 T^{3} + \)\(40\!\cdots\!16\)\( T^{4} + 93928718675848490 p^{7} T^{5} + 292914435841 p^{14} T^{6} + 389046 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 148778 T + 471562276492 T^{2} + 148715720905439662 T^{3} + \)\(10\!\cdots\!34\)\( T^{4} + 148715720905439662 p^{7} T^{5} + 471562276492 p^{14} T^{6} + 148778 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 154118 T + 714573950440 T^{2} - 59535070007086774 T^{3} + \)\(25\!\cdots\!42\)\( T^{4} - 59535070007086774 p^{7} T^{5} + 714573950440 p^{14} T^{6} - 154118 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 672234 T + 1025421513287 T^{2} - 582070988616741920 T^{3} + \)\(72\!\cdots\!60\)\( T^{4} - 582070988616741920 p^{7} T^{5} + 1025421513287 p^{14} T^{6} - 672234 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 1912606 T + 5428205441996 T^{2} + 127015644694927314 p T^{3} + \)\(99\!\cdots\!78\)\( T^{4} + 127015644694927314 p^{8} T^{5} + 5428205441996 p^{14} T^{6} + 1912606 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 865408 T + 2215428200531 T^{2} - 2921390178493833348 T^{3} + \)\(13\!\cdots\!72\)\( T^{4} - 2921390178493833348 p^{7} T^{5} + 2215428200531 p^{14} T^{6} - 865408 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 2204076 T + 12961764931012 T^{2} - 20450242149141569956 T^{3} + \)\(61\!\cdots\!22\)\( T^{4} - 20450242149141569956 p^{7} T^{5} + 12961764931012 p^{14} T^{6} - 2204076 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2867612 T + 9289642025747 T^{2} - 5326762945274333076 T^{3} - \)\(30\!\cdots\!24\)\( T^{4} - 5326762945274333076 p^{7} T^{5} + 9289642025747 p^{14} T^{6} + 2867612 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 9896448 T + 64517475818012 T^{2} + \)\(27\!\cdots\!32\)\( T^{3} + \)\(96\!\cdots\!54\)\( T^{4} + \)\(27\!\cdots\!32\)\( p^{7} T^{5} + 64517475818012 p^{14} T^{6} + 9896448 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1722988 T + 18967982783225 T^{2} + 33484475403143515926 T^{3} + \)\(32\!\cdots\!88\)\( T^{4} + 33484475403143515926 p^{7} T^{5} + 18967982783225 p^{14} T^{6} + 1722988 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 13037044 T + 118649444748076 T^{2} + \)\(77\!\cdots\!84\)\( T^{3} + \)\(38\!\cdots\!26\)\( T^{4} + \)\(77\!\cdots\!84\)\( p^{7} T^{5} + 118649444748076 p^{14} T^{6} + 13037044 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 7720032 T + 108366236633740 T^{2} + \)\(61\!\cdots\!08\)\( T^{3} + \)\(44\!\cdots\!22\)\( T^{4} + \)\(61\!\cdots\!08\)\( p^{7} T^{5} + 108366236633740 p^{14} T^{6} + 7720032 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 17191652 T + 215403147789956 T^{2} + \)\(19\!\cdots\!76\)\( T^{3} + \)\(15\!\cdots\!46\)\( T^{4} + \)\(19\!\cdots\!76\)\( p^{7} T^{5} + 215403147789956 p^{14} T^{6} + 17191652 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 4713534 T + 218551637402212 T^{2} + \)\(10\!\cdots\!62\)\( T^{3} + \)\(23\!\cdots\!54\)\( T^{4} + \)\(10\!\cdots\!62\)\( p^{7} T^{5} + 218551637402212 p^{14} T^{6} + 4713534 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.39866643477633767013192939360, −6.88935174947313499151030413707, −6.70132563005560367307818938862, −6.61494370782320622885914498042, −6.49977594241205170569173815281, −5.77273177525038169619355508922, −5.65675163761570302628981385524, −5.62855801979236333743612437328, −5.60452011422706048735333829387, −4.55469039573832289254325060143, −4.51575271076859982505114388083, −4.38790734423843758333734054847, −4.20362459474694996828796351700, −3.78511796468189479420584728406, −3.66234716947294711707054605367, −3.65430596492040464980209519609, −3.61376101020036973616717294444, −2.85948381977837278319335930730, −2.77471808091135881269048220037, −2.60932524333311980397964231111, −2.23100920771613169333529716858, −1.95753485537206121325085292543, −1.62448064998911903248146443493, −1.47516445654804243845859193764, −1.38004644954113734437977914427, 0, 0, 0, 0,
1.38004644954113734437977914427, 1.47516445654804243845859193764, 1.62448064998911903248146443493, 1.95753485537206121325085292543, 2.23100920771613169333529716858, 2.60932524333311980397964231111, 2.77471808091135881269048220037, 2.85948381977837278319335930730, 3.61376101020036973616717294444, 3.65430596492040464980209519609, 3.66234716947294711707054605367, 3.78511796468189479420584728406, 4.20362459474694996828796351700, 4.38790734423843758333734054847, 4.51575271076859982505114388083, 4.55469039573832289254325060143, 5.60452011422706048735333829387, 5.62855801979236333743612437328, 5.65675163761570302628981385524, 5.77273177525038169619355508922, 6.49977594241205170569173815281, 6.61494370782320622885914498042, 6.70132563005560367307818938862, 6.88935174947313499151030413707, 7.39866643477633767013192939360
