| L(s) = 1 | + 108·3-s − 196·5-s + 7.29e3·9-s − 406·11-s − 1.97e3·13-s − 2.11e4·15-s − 7.43e3·17-s + 1.58e4·19-s + 6.78e3·23-s − 1.71e5·25-s + 3.93e5·27-s − 9.45e4·29-s − 5.58e4·31-s − 4.38e4·33-s − 9.37e4·37-s − 2.13e5·39-s − 6.97e3·41-s − 2.43e5·43-s − 1.42e6·45-s − 6.50e5·47-s − 8.03e5·51-s − 3.68e5·53-s + 7.95e4·55-s + 1.71e6·57-s + 9.60e4·59-s − 3.61e6·61-s + 3.86e5·65-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.701·5-s + 10/3·9-s − 0.0919·11-s − 0.249·13-s − 1.61·15-s − 0.367·17-s + 0.530·19-s + 0.116·23-s − 2.20·25-s + 3.84·27-s − 0.719·29-s − 0.336·31-s − 0.212·33-s − 0.304·37-s − 0.575·39-s − 0.0157·41-s − 0.467·43-s − 2.33·45-s − 0.913·47-s − 0.847·51-s − 0.340·53-s + 0.0644·55-s + 1.22·57-s + 0.0608·59-s − 2.04·61-s + 0.174·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 196 T + 210407 T^{2} + 10166808 p T^{3} + 841615632 p^{2} T^{4} + 10166808 p^{8} T^{5} + 210407 p^{14} T^{6} + 196 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 406 T + 48797993 T^{2} + 18931376850 T^{3} + 1336309401103848 T^{4} + 18931376850 p^{7} T^{5} + 48797993 p^{14} T^{6} + 406 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 1974 T + 208939117 T^{2} + 412046927534 T^{3} + 18378156938146800 T^{4} + 412046927534 p^{7} T^{5} + 208939117 p^{14} T^{6} + 1974 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 7436 T + 1359824996 T^{2} + 8083602279012 T^{3} + 46810358167452198 p T^{4} + 8083602279012 p^{7} T^{5} + 1359824996 p^{14} T^{6} + 7436 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 15874 T + 2077600789 T^{2} + 1035457614938 T^{3} + 1804246086342007516 T^{4} + 1035457614938 p^{7} T^{5} + 2077600789 p^{14} T^{6} - 15874 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 6788 T + 4507787420 T^{2} - 38294904094932 T^{3} + 28311787602840204966 T^{4} - 38294904094932 p^{7} T^{5} + 4507787420 p^{14} T^{6} - 6788 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 94544 T + 39034925795 T^{2} + 6585126243044520 T^{3} + \)\(72\!\cdots\!16\)\( T^{4} + 6585126243044520 p^{7} T^{5} + 39034925795 p^{14} T^{6} + 94544 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 55890 T + 43582978972 T^{2} + 7702986651474680 T^{3} + \)\(11\!\cdots\!13\)\( T^{4} + 7702986651474680 p^{7} T^{5} + 43582978972 p^{14} T^{6} + 55890 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 93742 T + 22288997701 T^{2} - 2978638328215082 T^{3} + \)\(12\!\cdots\!88\)\( T^{4} - 2978638328215082 p^{7} T^{5} + 22288997701 p^{14} T^{6} + 93742 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 6972 T + 179204422376 T^{2} - 22687004651735916 T^{3} + \)\(68\!\cdots\!86\)\( T^{4} - 22687004651735916 p^{7} T^{5} + 179204422376 p^{14} T^{6} + 6972 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 243922 T + 668754369745 T^{2} + 172816838450048086 T^{3} + \)\(21\!\cdots\!36\)\( T^{4} + 172816838450048086 p^{7} T^{5} + 668754369745 p^{14} T^{6} + 243922 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 650484 T + 1467623883932 T^{2} + 954838747919275668 T^{3} + \)\(98\!\cdots\!50\)\( T^{4} + 954838747919275668 p^{7} T^{5} + 1467623883932 p^{14} T^{6} + 650484 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 6960 p T + 2593110423443 T^{2} + 1639733522959914480 T^{3} + \)\(37\!\cdots\!80\)\( T^{4} + 1639733522959914480 p^{7} T^{5} + 2593110423443 p^{14} T^{6} + 6960 p^{22} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 96026 T + 6929399410373 T^{2} - 3142914089992065786 T^{3} + \)\(21\!\cdots\!00\)\( T^{4} - 3142914089992065786 p^{7} T^{5} + 6929399410373 p^{14} T^{6} - 96026 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 3618156 T + 14412733384708 T^{2} + 32370824922494424548 T^{3} + \)\(71\!\cdots\!78\)\( T^{4} + 32370824922494424548 p^{7} T^{5} + 14412733384708 p^{14} T^{6} + 3618156 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 316006 T + 21643354119361 T^{2} - 3879809292566539982 T^{3} + \)\(18\!\cdots\!16\)\( T^{4} - 3879809292566539982 p^{7} T^{5} + 21643354119361 p^{14} T^{6} - 316006 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 6218156 T + 46176745736588 T^{2} - \)\(17\!\cdots\!76\)\( T^{3} + \)\(67\!\cdots\!62\)\( T^{4} - \)\(17\!\cdots\!76\)\( p^{7} T^{5} + 46176745736588 p^{14} T^{6} - 6218156 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1287286 T + 37990210228537 T^{2} + 42641123322990781238 T^{3} + \)\(59\!\cdots\!84\)\( T^{4} + 42641123322990781238 p^{7} T^{5} + 37990210228537 p^{14} T^{6} + 1287286 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 8187282 T + 71605815972892 T^{2} + \)\(45\!\cdots\!68\)\( T^{3} + \)\(20\!\cdots\!37\)\( T^{4} + \)\(45\!\cdots\!68\)\( p^{7} T^{5} + 71605815972892 p^{14} T^{6} + 8187282 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 3693650 T + 66867395206505 T^{2} + \)\(20\!\cdots\!94\)\( T^{3} + \)\(22\!\cdots\!04\)\( T^{4} + \)\(20\!\cdots\!94\)\( p^{7} T^{5} + 66867395206505 p^{14} T^{6} + 3693650 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 14489928 T + 79108388607632 T^{2} - 89332237812223923624 T^{3} - \)\(31\!\cdots\!18\)\( T^{4} - 89332237812223923624 p^{7} T^{5} + 79108388607632 p^{14} T^{6} + 14489928 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 11861290 T + 293699437376665 T^{2} + \)\(23\!\cdots\!50\)\( T^{3} + \)\(33\!\cdots\!96\)\( T^{4} + \)\(23\!\cdots\!50\)\( p^{7} T^{5} + 293699437376665 p^{14} T^{6} + 11861290 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.40502274497656227281806897604, −6.80875807994771310536420809108, −6.55387798117867356003454257407, −6.50086887077164431370007738161, −6.30068346983199856671908735405, −5.69670298816320717003582055594, −5.49238297763557497701300039195, −5.34000023508638353624018265542, −5.22528727393304311646607662293, −4.61283022047249991940620554416, −4.34637821189083153009995395528, −4.22891303861983973417539323576, −4.22158975583988921144439245403, −3.56316515435135889150165610359, −3.55725117573849011244469709347, −3.42218078673459530523953764861, −3.15956253462106695317055311219, −2.62593994327689382564445830086, −2.50194040981503601843117367103, −2.31102001834525998471815361512, −2.16127503486407707009019211769, −1.51802222585711472460766791396, −1.35395498769185484508126362680, −1.34437921513780327570842031673, −1.06380777419738511475784647893, 0, 0, 0, 0,
1.06380777419738511475784647893, 1.34437921513780327570842031673, 1.35395498769185484508126362680, 1.51802222585711472460766791396, 2.16127503486407707009019211769, 2.31102001834525998471815361512, 2.50194040981503601843117367103, 2.62593994327689382564445830086, 3.15956253462106695317055311219, 3.42218078673459530523953764861, 3.55725117573849011244469709347, 3.56316515435135889150165610359, 4.22158975583988921144439245403, 4.22891303861983973417539323576, 4.34637821189083153009995395528, 4.61283022047249991940620554416, 5.22528727393304311646607662293, 5.34000023508638353624018265542, 5.49238297763557497701300039195, 5.69670298816320717003582055594, 6.30068346983199856671908735405, 6.50086887077164431370007738161, 6.55387798117867356003454257407, 6.80875807994771310536420809108, 7.40502274497656227281806897604
