| L(s) = 1 | + 16·2-s − 60·4-s − 2.97e3·8-s + 834·9-s − 3.76e3·11-s − 1.22e4·16-s + 1.33e4·18-s − 6.02e4·22-s − 4.94e3·23-s + 2.87e4·25-s − 3.45e4·29-s + 2.40e5·32-s − 5.00e4·36-s + 8.77e4·37-s + 3.20e5·43-s + 2.25e5·44-s − 7.90e4·46-s + 4.60e5·50-s − 2.09e5·53-s − 5.52e5·58-s + 2.42e6·64-s − 8.64e5·67-s − 1.48e6·71-s − 2.48e6·72-s + 1.40e6·74-s − 1.87e6·79-s − 4.31e5·81-s + ⋯ |
| L(s) = 1 | + 2·2-s − 0.937·4-s − 5.81·8-s + 1.14·9-s − 2.82·11-s − 2.99·16-s + 2.28·18-s − 5.65·22-s − 0.406·23-s + 1.83·25-s − 1.41·29-s + 7.32·32-s − 1.07·36-s + 1.73·37-s + 4.02·43-s + 2.65·44-s − 0.812·46-s + 3.67·50-s − 1.40·53-s − 2.83·58-s + 9.25·64-s − 2.87·67-s − 4.13·71-s − 6.64·72-s + 3.46·74-s − 3.79·79-s − 0.812·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5764801 ^{s/2} \, \Gamma_{\C}(s+3)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.788885424\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.788885424\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( ( 1 - p^{3} T + 63 p T^{2} - p^{9} T^{3} + p^{12} T^{4} )^{2} \) |
| 3 | $D_4\times C_2$ | \( 1 - 278 p T^{2} + 125251 p^{2} T^{4} - 278 p^{13} T^{6} + p^{24} T^{8} \) |
| 5 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 p^{2} T - 21 p^{4} T^{2} - 2 p^{8} T^{3} + p^{12} T^{4} )( 1 + 2 p^{2} T - 21 p^{4} T^{2} + 2 p^{8} T^{3} + p^{12} T^{4} ) \) |
| 11 | $D_{4}$ | \( ( 1 + 1882 T + 3838905 T^{2} + 1882 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 5662108 T^{2} + 45798091584678 T^{4} - 5662108 p^{12} T^{6} + p^{24} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 66290974 T^{2} + 2261663941974339 T^{4} - 66290974 p^{12} T^{6} + p^{24} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 132859042 T^{2} + 8792914599953211 T^{4} - 132859042 p^{12} T^{6} + p^{24} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 2470 T + 85705305 T^{2} + 2470 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 17272 T + 1034818938 T^{2} + 17272 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 3401922658 T^{2} + 4463723664316462875 T^{4} - 3401922658 p^{12} T^{6} + p^{24} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 43870 T + 4906318515 T^{2} - 43870 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 17928625852 T^{2} + \)\(12\!\cdots\!86\)\( T^{4} - 17928625852 p^{12} T^{6} + p^{24} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 160108 T + 18917216814 T^{2} - 160108 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 19263896338 T^{2} + \)\(26\!\cdots\!51\)\( T^{4} - 19263896338 p^{12} T^{6} + p^{24} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 104530 T + 6310937283 T^{2} + 104530 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 155549342866 T^{2} + \)\(95\!\cdots\!19\)\( T^{4} - 155549342866 p^{12} T^{6} + p^{24} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 196054017790 T^{2} + \)\(14\!\cdots\!67\)\( T^{4} - 196054017790 p^{12} T^{6} + p^{24} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 432494 T + 218546768097 T^{2} + 432494 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 740764 T + 392433088158 T^{2} + 740764 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 140806261870 T^{2} + \)\(24\!\cdots\!59\)\( T^{4} - 140806261870 p^{12} T^{6} + p^{24} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 935990 T + 683336589705 T^{2} + 935990 p^{6} T^{3} + p^{12} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 467743512772 T^{2} + \)\(19\!\cdots\!90\)\( T^{4} - 467743512772 p^{12} T^{6} + p^{24} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 1248409583422 T^{2} + \)\(83\!\cdots\!63\)\( T^{4} - 1248409583422 p^{12} T^{6} + p^{24} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 2133147299644 T^{2} + \)\(22\!\cdots\!54\)\( T^{4} - 2133147299644 p^{12} T^{6} + p^{24} T^{8} \) |
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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
−10.26852132506750912791980131167, −10.15904123487134995532319521680, −9.636875283352707368636918952200, −9.385977660868910288207110333524, −8.986511393728475466669482266752, −8.749214729272753463498394680270, −8.588257443491028571011194660989, −7.87422175768372870781949121965, −7.57186806853564217499639497147, −7.50671660623642406040306689109, −6.98312834902249422565613045275, −5.86857368351518607262357309478, −5.86345067885781950515643183828, −5.79932581400204506909103928028, −5.31010751931213115390759913299, −4.75009260802143116111713056445, −4.35486617893283502644764589813, −4.34855480792709910689693432712, −4.27805006486738202632136614221, −3.18984031123981765436261053561, −2.91286435839154014186397358143, −2.81081303314465135137337813311, −1.62439285474341168921535722388, −0.65181894478424413101430559891, −0.31889798821356660037206836154,
0.31889798821356660037206836154, 0.65181894478424413101430559891, 1.62439285474341168921535722388, 2.81081303314465135137337813311, 2.91286435839154014186397358143, 3.18984031123981765436261053561, 4.27805006486738202632136614221, 4.34855480792709910689693432712, 4.35486617893283502644764589813, 4.75009260802143116111713056445, 5.31010751931213115390759913299, 5.79932581400204506909103928028, 5.86345067885781950515643183828, 5.86857368351518607262357309478, 6.98312834902249422565613045275, 7.50671660623642406040306689109, 7.57186806853564217499639497147, 7.87422175768372870781949121965, 8.588257443491028571011194660989, 8.749214729272753463498394680270, 8.986511393728475466669482266752, 9.385977660868910288207110333524, 9.636875283352707368636918952200, 10.15904123487134995532319521680, 10.26852132506750912791980131167
