| L(s) = 1 | − 15·2-s + 139·4-s − 306·5-s + 890·7-s − 1.13e3·8-s + 4.59e3·10-s − 5.32e3·11-s − 1.82e3·13-s − 1.33e4·14-s + 6.30e3·16-s − 3.28e4·17-s − 1.27e4·19-s − 4.25e4·20-s + 7.98e4·22-s − 1.14e5·23-s − 7.30e4·25-s + 2.73e4·26-s + 1.23e5·28-s + 1.04e5·29-s − 2.49e4·31-s − 2.87e4·32-s + 4.92e5·34-s − 2.72e5·35-s − 4.98e5·37-s + 1.91e5·38-s + 3.48e5·40-s − 7.34e5·41-s + ⋯ |
| L(s) = 1 | − 1.32·2-s + 1.08·4-s − 1.09·5-s + 0.980·7-s − 0.786·8-s + 1.45·10-s − 1.20·11-s − 0.230·13-s − 1.30·14-s + 0.384·16-s − 1.62·17-s − 0.427·19-s − 1.18·20-s + 1.59·22-s − 1.96·23-s − 0.934·25-s + 0.304·26-s + 1.06·28-s + 0.799·29-s − 0.150·31-s − 0.155·32-s + 2.15·34-s − 1.07·35-s − 1.61·37-s + 0.566·38-s + 0.861·40-s − 1.66·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96059601 ^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | | \( 1 \) |
| 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
−9.683491495356719184855626064883, −8.774879711264805856454578483850, −8.636669871144692277549959880886, −8.497099498838615298773336764512, −8.344820658854856379023722132268, −7.85335003320488045400802429182, −7.77974138442013602546470118045, −7.69405564633026521829936464026, −6.98263870560572429759542214832, −6.77334985801154164989925109788, −6.66340446337099534293482698553, −6.07067395572391909117083682717, −5.91081719300275434390285740094, −5.31171342641242082068851498771, −4.83685332582601139277219229438, −4.75965155861440804605976923978, −4.53974387770926403040788266446, −3.73868815652103181527046903095, −3.67557779798610400412394244425, −3.25202754697679852925386096635, −2.55150399681140662816300194580, −2.31251783809451923830956311775, −1.72507395327250519896343012619, −1.71678583285675976554979631178, −1.18682916133191141120369281167, 0, 0, 0, 0,
1.18682916133191141120369281167, 1.71678583285675976554979631178, 1.72507395327250519896343012619, 2.31251783809451923830956311775, 2.55150399681140662816300194580, 3.25202754697679852925386096635, 3.67557779798610400412394244425, 3.73868815652103181527046903095, 4.53974387770926403040788266446, 4.75965155861440804605976923978, 4.83685332582601139277219229438, 5.31171342641242082068851498771, 5.91081719300275434390285740094, 6.07067395572391909117083682717, 6.66340446337099534293482698553, 6.77334985801154164989925109788, 6.98263870560572429759542214832, 7.69405564633026521829936464026, 7.77974138442013602546470118045, 7.85335003320488045400802429182, 8.344820658854856379023722132268, 8.497099498838615298773336764512, 8.636669871144692277549959880886, 8.774879711264805856454578483850, 9.683491495356719184855626064883
