| L(s) = 1 | − 10·5-s − 14·7-s + 52·13-s − 26·17-s − 28·19-s + 164·23-s − 111·25-s − 174·29-s − 318·31-s + 140·35-s + 296·37-s + 118·41-s + 260·43-s + 204·47-s − 253·49-s − 1.08e3·53-s + 196·59-s + 1.53e3·61-s − 520·65-s + 660·67-s − 852·71-s − 478·73-s − 22·79-s + 1.13e3·83-s + 260·85-s − 110·89-s − 728·91-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.755·7-s + 1.10·13-s − 0.370·17-s − 0.338·19-s + 1.48·23-s − 0.887·25-s − 1.11·29-s − 1.84·31-s + 0.676·35-s + 1.31·37-s + 0.449·41-s + 0.922·43-s + 0.633·47-s − 0.737·49-s − 2.81·53-s + 0.432·59-s + 3.22·61-s − 0.992·65-s + 1.20·67-s − 1.42·71-s − 0.766·73-s − 0.0313·79-s + 1.50·83-s + 0.331·85-s − 0.131·89-s − 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{3} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 2 p T + 211 T^{2} + 2396 T^{3} + 211 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 7 | $S_4\times C_2$ | \( 1 + 2 p T + 449 T^{2} + 3788 T^{3} + 449 p^{3} T^{4} + 2 p^{7} T^{5} + p^{9} T^{6} \) |
| 11 | $S_4\times C_2$ | \( 1 + 107 p T^{2} + 49152 T^{3} + 107 p^{4} T^{4} + p^{9} T^{6} \) |
| 13 | $S_4\times C_2$ | \( 1 - 4 p T + 5487 T^{2} - 172616 T^{3} + 5487 p^{3} T^{4} - 4 p^{7} T^{5} + p^{9} T^{6} \) |
| 17 | $S_4\times C_2$ | \( 1 + 26 T + 3615 T^{2} - 222100 T^{3} + 3615 p^{3} T^{4} + 26 p^{6} T^{5} + p^{9} T^{6} \) |
| 19 | $S_4\times C_2$ | \( 1 + 28 T + 9489 T^{2} + 658856 T^{3} + 9489 p^{3} T^{4} + 28 p^{6} T^{5} + p^{9} T^{6} \) |
| 23 | $S_4\times C_2$ | \( 1 - 164 T + 42885 T^{2} - 4036280 T^{3} + 42885 p^{3} T^{4} - 164 p^{6} T^{5} + p^{9} T^{6} \) |
| 29 | $S_4\times C_2$ | \( 1 + 6 p T + 77131 T^{2} + 8426004 T^{3} + 77131 p^{3} T^{4} + 6 p^{7} T^{5} + p^{9} T^{6} \) |
| 31 | $S_4\times C_2$ | \( 1 + 318 T + 93849 T^{2} + 15197452 T^{3} + 93849 p^{3} T^{4} + 318 p^{6} T^{5} + p^{9} T^{6} \) |
| 37 | $S_4\times C_2$ | \( 1 - 8 p T + 105879 T^{2} - 29903632 T^{3} + 105879 p^{3} T^{4} - 8 p^{7} T^{5} + p^{9} T^{6} \) |
| 41 | $S_4\times C_2$ | \( 1 - 118 T + 89463 T^{2} + 3720620 T^{3} + 89463 p^{3} T^{4} - 118 p^{6} T^{5} + p^{9} T^{6} \) |
| 43 | $S_4\times C_2$ | \( 1 - 260 T + 157545 T^{2} - 32742232 T^{3} + 157545 p^{3} T^{4} - 260 p^{6} T^{5} + p^{9} T^{6} \) |
| 47 | $S_4\times C_2$ | \( 1 - 204 T + 283677 T^{2} - 40395048 T^{3} + 283677 p^{3} T^{4} - 204 p^{6} T^{5} + p^{9} T^{6} \) |
| 53 | $S_4\times C_2$ | \( 1 + 1086 T + 736099 T^{2} + 344026068 T^{3} + 736099 p^{3} T^{4} + 1086 p^{6} T^{5} + p^{9} T^{6} \) |
| 59 | $S_4\times C_2$ | \( 1 - 196 T + 566137 T^{2} - 71984984 T^{3} + 566137 p^{3} T^{4} - 196 p^{6} T^{5} + p^{9} T^{6} \) |
| 61 | $S_4\times C_2$ | \( 1 - 1536 T + 1409583 T^{2} - 802126848 T^{3} + 1409583 p^{3} T^{4} - 1536 p^{6} T^{5} + p^{9} T^{6} \) |
| 67 | $S_4\times C_2$ | \( 1 - 660 T + 592833 T^{2} - 364778232 T^{3} + 592833 p^{3} T^{4} - 660 p^{6} T^{5} + p^{9} T^{6} \) |
| 71 | $S_4\times C_2$ | \( 1 + 12 p T + 1006773 T^{2} + 524795352 T^{3} + 1006773 p^{3} T^{4} + 12 p^{7} T^{5} + p^{9} T^{6} \) |
| 73 | $S_4\times C_2$ | \( 1 + 478 T + 911095 T^{2} + 251066948 T^{3} + 911095 p^{3} T^{4} + 478 p^{6} T^{5} + p^{9} T^{6} \) |
| 79 | $S_4\times C_2$ | \( 1 + 22 T + 1407593 T^{2} + 13791100 T^{3} + 1407593 p^{3} T^{4} + 22 p^{6} T^{5} + p^{9} T^{6} \) |
| 83 | $S_4\times C_2$ | \( 1 - 1136 T + 2100385 T^{2} - 1337053600 T^{3} + 2100385 p^{3} T^{4} - 1136 p^{6} T^{5} + p^{9} T^{6} \) |
| 89 | $S_4\times C_2$ | \( 1 + 110 T + 2073543 T^{2} + 156516836 T^{3} + 2073543 p^{3} T^{4} + 110 p^{6} T^{5} + p^{9} T^{6} \) |
| 97 | $S_4\times C_2$ | \( 1 + 1222 T + 2989679 T^{2} + 2155770388 T^{3} + 2989679 p^{3} T^{4} + 1222 p^{6} T^{5} + p^{9} T^{6} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{6} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.174867438515507912210288038799, −7.71899682805336047532558857828, −7.41232439831194670696000273754, −7.34526144841636949815537916390, −7.00284062929883372113606701619, −6.80440000257434504141702737302, −6.41968218021287014293133618087, −6.12791957517825065652265645149, −6.04135799305823506422250273549, −5.70547508128664362105545376339, −5.33056899450927704920521415740, −5.14557511045989853502088775702, −4.83281176297986454278127786194, −4.34325743972343366327447565094, −4.12958835811264109988466107391, −3.96037047807753391509614626585, −3.50006557313923979813306503504, −3.40438550102475675079778359267, −3.27715232715275875100943533310, −2.48461754772590579812449203653, −2.32086857435270799238705390177, −2.24082001846737911446394193538, −1.31982674604136749808391830549, −1.26375981333928857382622728934, −0.955713148655077097087410123678, 0, 0, 0,
0.955713148655077097087410123678, 1.26375981333928857382622728934, 1.31982674604136749808391830549, 2.24082001846737911446394193538, 2.32086857435270799238705390177, 2.48461754772590579812449203653, 3.27715232715275875100943533310, 3.40438550102475675079778359267, 3.50006557313923979813306503504, 3.96037047807753391509614626585, 4.12958835811264109988466107391, 4.34325743972343366327447565094, 4.83281176297986454278127786194, 5.14557511045989853502088775702, 5.33056899450927704920521415740, 5.70547508128664362105545376339, 6.04135799305823506422250273549, 6.12791957517825065652265645149, 6.41968218021287014293133618087, 6.80440000257434504141702737302, 7.00284062929883372113606701619, 7.34526144841636949815537916390, 7.41232439831194670696000273754, 7.71899682805336047532558857828, 8.174867438515507912210288038799
