| L(s) = 1 | + 108·3-s + 192·5-s + 680·7-s + 7.29e3·9-s − 4.49e3·11-s − 1.28e4·13-s + 2.07e4·15-s − 1.49e4·17-s − 2.15e4·19-s + 7.34e4·21-s + 5.49e4·23-s − 4.24e4·25-s + 3.93e5·27-s + 2.42e5·29-s + 1.51e5·31-s − 4.85e5·33-s + 1.30e5·35-s + 1.13e5·37-s − 1.38e6·39-s − 2.39e5·41-s + 1.49e6·43-s + 1.39e6·45-s + 7.72e5·47-s − 2.20e6·49-s − 1.61e6·51-s + 2.38e6·53-s − 8.63e5·55-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.686·5-s + 0.749·7-s + 10/3·9-s − 1.01·11-s − 1.62·13-s + 1.58·15-s − 0.738·17-s − 0.719·19-s + 1.73·21-s + 0.942·23-s − 0.543·25-s + 3.84·27-s + 1.84·29-s + 0.912·31-s − 2.35·33-s + 0.514·35-s + 0.367·37-s − 3.74·39-s − 0.541·41-s + 2.86·43-s + 2.28·45-s + 1.08·47-s − 2.67·49-s − 1.70·51-s + 2.20·53-s − 0.699·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(25.27008242\) |
| \(L(\frac12)\) |
\(\approx\) |
\(25.27008242\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 192 T + 79316 T^{2} - 4433984 p T^{3} + 126774822 p^{2} T^{4} - 4433984 p^{8} T^{5} + 79316 p^{14} T^{6} - 192 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 680 T + 2666836 T^{2} - 1272950888 T^{3} + 2998744001542 T^{4} - 1272950888 p^{7} T^{5} + 2666836 p^{14} T^{6} - 680 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 4496 T + 26191340 T^{2} + 181969297168 T^{3} + 813341110124918 T^{4} + 181969297168 p^{7} T^{5} + 26191340 p^{14} T^{6} + 4496 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 12840 T + 132115948 T^{2} + 713027073272 T^{3} + 6498619482769302 T^{4} + 713027073272 p^{7} T^{5} + 132115948 p^{14} T^{6} + 12840 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 14952 T + 113915420 T^{2} + 5793309697112 T^{3} + 258267405954490374 T^{4} + 5793309697112 p^{7} T^{5} + 113915420 p^{14} T^{6} + 14952 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 21504 T + 2446733932 T^{2} + 53249972851712 T^{3} + 2887387256916153750 T^{4} + 53249972851712 p^{7} T^{5} + 2446733932 p^{14} T^{6} + 21504 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 54992 T + 1516634492 T^{2} + 249648798850544 T^{3} - 23815362010402645018 T^{4} + 249648798850544 p^{7} T^{5} + 1516634492 p^{14} T^{6} - 54992 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 242384 T + 34768900820 T^{2} - 146365951614192 p T^{3} + \)\(60\!\cdots\!98\)\( T^{4} - 146365951614192 p^{8} T^{5} + 34768900820 p^{14} T^{6} - 242384 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 151432 T + 93181101748 T^{2} - 9876646593730824 T^{3} + \)\(35\!\cdots\!90\)\( T^{4} - 9876646593730824 p^{7} T^{5} + 93181101748 p^{14} T^{6} - 151432 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 113288 T + 266022576076 T^{2} - 28745603031036632 T^{3} + \)\(34\!\cdots\!18\)\( T^{4} - 28745603031036632 p^{7} T^{5} + 266022576076 p^{14} T^{6} - 113288 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 239176 T + 326935809404 T^{2} + 207369357289681464 T^{3} + \)\(56\!\cdots\!02\)\( T^{4} + 207369357289681464 p^{7} T^{5} + 326935809404 p^{14} T^{6} + 239176 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1495328 T + 1369179794572 T^{2} - 952507717741307424 T^{3} + \)\(55\!\cdots\!42\)\( T^{4} - 952507717741307424 p^{7} T^{5} + 1369179794572 p^{14} T^{6} - 1495328 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 772368 T + 415239683612 T^{2} - 145238618306648272 T^{3} + \)\(28\!\cdots\!66\)\( T^{4} - 145238618306648272 p^{7} T^{5} + 415239683612 p^{14} T^{6} - 772368 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 2389776 T + 6682144612724 T^{2} - 9100205902823940720 T^{3} + \)\(13\!\cdots\!14\)\( T^{4} - 9100205902823940720 p^{7} T^{5} + 6682144612724 p^{14} T^{6} - 2389776 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 141232 T + 4417645030316 T^{2} - 4475890865907499312 T^{3} + \)\(94\!\cdots\!02\)\( T^{4} - 4475890865907499312 p^{7} T^{5} + 4417645030316 p^{14} T^{6} - 141232 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 1231304 T + 7316522672620 T^{2} - 136484534453100856 p T^{3} + \)\(33\!\cdots\!14\)\( T^{4} - 136484534453100856 p^{8} T^{5} + 7316522672620 p^{14} T^{6} - 1231304 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 441392 T + 18348417993868 T^{2} + 6841337925338579248 T^{3} + \)\(15\!\cdots\!74\)\( T^{4} + 6841337925338579248 p^{7} T^{5} + 18348417993868 p^{14} T^{6} + 441392 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 1507504 T + 18085310739644 T^{2} - 20591960554472314992 T^{3} + \)\(21\!\cdots\!18\)\( T^{4} - 20591960554472314992 p^{7} T^{5} + 18085310739644 p^{14} T^{6} - 1507504 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1516840 T + 33953560223548 T^{2} + 24308897939228417368 T^{3} + \)\(49\!\cdots\!42\)\( T^{4} + 24308897939228417368 p^{7} T^{5} + 33953560223548 p^{14} T^{6} + 1516840 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 9540936 T + 50943310648948 T^{2} - 79126620358128821320 T^{3} + \)\(92\!\cdots\!54\)\( T^{4} - 79126620358128821320 p^{7} T^{5} + 50943310648948 p^{14} T^{6} - 9540936 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 4587600 T + 71876902198988 T^{2} + \)\(26\!\cdots\!56\)\( T^{3} + \)\(24\!\cdots\!02\)\( T^{4} + \)\(26\!\cdots\!56\)\( p^{7} T^{5} + 71876902198988 p^{14} T^{6} + 4587600 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 162376 T + 117629517059132 T^{2} + \)\(20\!\cdots\!52\)\( T^{3} + \)\(62\!\cdots\!70\)\( T^{4} + \)\(20\!\cdots\!52\)\( p^{7} T^{5} + 117629517059132 p^{14} T^{6} - 162376 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 2726760 T + 194690723817052 T^{2} - \)\(63\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!14\)\( T^{4} - \)\(63\!\cdots\!40\)\( p^{7} T^{5} + 194690723817052 p^{14} T^{6} - 2726760 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−7.23863727001184823005031672908, −6.81602393966748388881256814089, −6.42744691369389915850605404262, −6.42578287853184756710345704909, −6.20031304311768936214002131339, −5.52926914716258696109420377027, −5.33171183833560836054143137339, −5.25070925563246375859317594200, −4.85975831524861752069157083430, −4.49043105620446623285615228701, −4.40156698091464796058795630452, −4.15878504538515884473032848267, −3.99612152191859405636831761355, −3.34742435004788176262483216696, −3.07335338671218980505697670137, −2.87919142671959377814331967701, −2.75343699660312644344482331652, −2.26757520242856909662488006140, −2.20588135183470170732372068306, −1.97304497935835854052828255727, −1.82265125171952310098522853670, −1.14108510848980635296922093636, −0.913151859014131801593093573727, −0.64258095646192352351377159147, −0.33096069031303393593107961324,
0.33096069031303393593107961324, 0.64258095646192352351377159147, 0.913151859014131801593093573727, 1.14108510848980635296922093636, 1.82265125171952310098522853670, 1.97304497935835854052828255727, 2.20588135183470170732372068306, 2.26757520242856909662488006140, 2.75343699660312644344482331652, 2.87919142671959377814331967701, 3.07335338671218980505697670137, 3.34742435004788176262483216696, 3.99612152191859405636831761355, 4.15878504538515884473032848267, 4.40156698091464796058795630452, 4.49043105620446623285615228701, 4.85975831524861752069157083430, 5.25070925563246375859317594200, 5.33171183833560836054143137339, 5.52926914716258696109420377027, 6.20031304311768936214002131339, 6.42578287853184756710345704909, 6.42744691369389915850605404262, 6.81602393966748388881256814089, 7.23863727001184823005031672908
