| L(s) = 1 | + 15·2-s + 108·3-s + 139·4-s + 306·5-s + 1.62e3·6-s + 890·7-s + 1.13e3·8-s + 7.29e3·9-s + 4.59e3·10-s + 5.32e3·11-s + 1.50e4·12-s − 1.82e3·13-s + 1.33e4·14-s + 3.30e4·15-s + 6.30e3·16-s + 3.28e4·17-s + 1.09e5·18-s − 1.27e4·19-s + 4.25e4·20-s + 9.61e4·21-s + 7.98e4·22-s + 1.14e5·23-s + 1.23e5·24-s − 7.30e4·25-s − 2.73e4·26-s + 3.93e5·27-s + 1.23e5·28-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 2.30·3-s + 1.08·4-s + 1.09·5-s + 3.06·6-s + 0.980·7-s + 0.786·8-s + 10/3·9-s + 1.45·10-s + 1.20·11-s + 2.50·12-s − 0.230·13-s + 1.30·14-s + 2.52·15-s + 0.384·16-s + 1.62·17-s + 4.41·18-s − 0.427·19-s + 1.18·20-s + 2.26·21-s + 1.59·22-s + 1.96·23-s + 1.81·24-s − 0.934·25-s − 0.304·26-s + 3.84·27-s + 1.06·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(48.46046306\) |
| \(L(\frac12)\) |
\(\approx\) |
\(48.46046306\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - 15 T + 43 p T^{2} - 43 p^{3} T^{3} + 249 p^{4} T^{4} - 43 p^{10} T^{5} + 43 p^{15} T^{6} - 15 p^{21} T^{7} + p^{28} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 306 T + 33328 p T^{2} - 1192838 p^{2} T^{3} + 120655398 p^{3} T^{4} - 1192838 p^{9} T^{5} + 33328 p^{15} T^{6} - 306 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 890 T + 2428648 T^{2} - 1513098682 T^{3} + 2713357113806 T^{4} - 1513098682 p^{7} T^{5} + 2428648 p^{14} T^{6} - 890 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 1822 T + 143434048 T^{2} + 57178087826 p T^{3} + 9963947917232846 T^{4} + 57178087826 p^{8} T^{5} + 143434048 p^{14} T^{6} + 1822 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 32856 T + 1617694640 T^{2} - 2092957427944 p T^{3} + 1006917500477444958 T^{4} - 2092957427944 p^{8} T^{5} + 1617694640 p^{14} T^{6} - 32856 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 12784 T + 1968859072 T^{2} + 47795746564128 T^{3} + 103428707615650746 p T^{4} + 47795746564128 p^{7} T^{5} + 1968859072 p^{14} T^{6} + 12784 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 114858 T + 9605711756 T^{2} - 598055876188978 T^{3} + 37322607621124682694 T^{4} - 598055876188978 p^{7} T^{5} + 9605711756 p^{14} T^{6} - 114858 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 104952 T + 52207854272 T^{2} + 4176821606180040 T^{3} + \)\(12\!\cdots\!34\)\( T^{4} + 4176821606180040 p^{7} T^{5} + 52207854272 p^{14} T^{6} + 104952 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 24976 T + 43688791996 T^{2} + 2739115850747216 T^{3} + \)\(14\!\cdots\!22\)\( T^{4} + 2739115850747216 p^{7} T^{5} + 43688791996 p^{14} T^{6} + 24976 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 498856 T + 381977216092 T^{2} + 132519305800020216 T^{3} + \)\(54\!\cdots\!66\)\( T^{4} + 132519305800020216 p^{7} T^{5} + 381977216092 p^{14} T^{6} + 498856 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 17916 p T + 582543752696 T^{2} - 174072649937861684 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} - 174072649937861684 p^{7} T^{5} + 582543752696 p^{14} T^{6} - 17916 p^{22} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 201916 T + 337036845088 T^{2} - 169098883219921380 T^{3} + \)\(16\!\cdots\!34\)\( T^{4} - 169098883219921380 p^{7} T^{5} + 337036845088 p^{14} T^{6} + 201916 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 1995894 T + 2541886433372 T^{2} - 2356429844123408062 T^{3} + \)\(18\!\cdots\!58\)\( T^{4} - 2356429844123408062 p^{7} T^{5} + 2541886433372 p^{14} T^{6} - 1995894 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 929970 T + 696841347632 T^{2} + 410483380995330858 T^{3} - \)\(19\!\cdots\!26\)\( T^{4} + 410483380995330858 p^{7} T^{5} + 696841347632 p^{14} T^{6} - 929970 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1353156 T + 641417092700 T^{2} - 638087859099231076 T^{3} + \)\(35\!\cdots\!18\)\( T^{4} - 638087859099231076 p^{7} T^{5} + 641417092700 p^{14} T^{6} - 1353156 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 3998774 T + 16202477625640 T^{2} - 36808547102293149042 T^{3} + \)\(80\!\cdots\!50\)\( T^{4} - 36808547102293149042 p^{7} T^{5} + 16202477625640 p^{14} T^{6} - 3998774 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1722008 T + 22644499591660 T^{2} - 31607526991944054296 T^{3} + \)\(20\!\cdots\!06\)\( T^{4} - 31607526991944054296 p^{7} T^{5} + 22644499591660 p^{14} T^{6} - 1722008 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 5571858 T + 36809238259148 T^{2} - \)\(11\!\cdots\!98\)\( T^{3} + \)\(45\!\cdots\!94\)\( T^{4} - \)\(11\!\cdots\!98\)\( p^{7} T^{5} + 36809238259148 p^{14} T^{6} - 5571858 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 5600528 T + 40596655684732 T^{2} - \)\(18\!\cdots\!84\)\( T^{3} + \)\(65\!\cdots\!02\)\( T^{4} - \)\(18\!\cdots\!84\)\( p^{7} T^{5} + 40596655684732 p^{14} T^{6} - 5600528 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 7710226 T + 92181525864424 T^{2} + \)\(44\!\cdots\!70\)\( T^{3} + \)\(27\!\cdots\!30\)\( T^{4} + \)\(44\!\cdots\!70\)\( p^{7} T^{5} + 92181525864424 p^{14} T^{6} + 7710226 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 3431856 T + 43757629855388 T^{2} - 89436476395110295728 T^{3} + \)\(91\!\cdots\!66\)\( T^{4} - 89436476395110295728 p^{7} T^{5} + 43757629855388 p^{14} T^{6} - 3431856 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 4611528 T + 150275359100156 T^{2} - \)\(46\!\cdots\!64\)\( T^{3} + \)\(92\!\cdots\!06\)\( T^{4} - \)\(46\!\cdots\!64\)\( p^{7} T^{5} + 150275359100156 p^{14} T^{6} - 4611528 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 1401692 T + 169506769846084 T^{2} - \)\(11\!\cdots\!24\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!24\)\( p^{7} T^{5} + 169506769846084 p^{14} T^{6} - 1401692 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−10.83730530230477760731544375658, −10.74834237437520131266032869389, −10.06874079327326950994687170545, −9.813383602497820916257371635871, −9.608229722204437014445829817084, −9.153107922119312846568116185085, −8.807707209032929220632494108771, −8.658741435693496609090898333078, −8.064791962740171527177018692022, −7.61912587593214194642885544873, −7.51919985911852597013658284710, −6.99370953774994314075886551768, −6.57951661563364711818909767817, −6.16641085540008548894515054368, −5.40101921208311571535378750923, −5.24561324460413851143716351903, −4.85668052090962526995394042924, −3.98697527369460417736824308619, −3.69475491268798158751465551929, −3.68826716226350003195080833812, −2.75464643051211002311515262324, −2.30622516399069560920816987564, −2.06964773939259369133903744582, −1.23643478649147673765147532443, −1.14283042662437233346125214717,
1.14283042662437233346125214717, 1.23643478649147673765147532443, 2.06964773939259369133903744582, 2.30622516399069560920816987564, 2.75464643051211002311515262324, 3.68826716226350003195080833812, 3.69475491268798158751465551929, 3.98697527369460417736824308619, 4.85668052090962526995394042924, 5.24561324460413851143716351903, 5.40101921208311571535378750923, 6.16641085540008548894515054368, 6.57951661563364711818909767817, 6.99370953774994314075886551768, 7.51919985911852597013658284710, 7.61912587593214194642885544873, 8.064791962740171527177018692022, 8.658741435693496609090898333078, 8.807707209032929220632494108771, 9.153107922119312846568116185085, 9.608229722204437014445829817084, 9.813383602497820916257371635871, 10.06874079327326950994687170545, 10.74834237437520131266032869389, 10.83730530230477760731544375658
