| 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
−8.030114330952035488021400952348, −7.51920276436931709981823479390, −7.39613262024394233978612992189, −7.32621853969158218206902649164, −6.93869053722156150634991688501, −6.92068287342766729979839919191, −6.31583635588837666740889752562, −6.21041422458127661637363621755, −5.87110118822486464694552313330, −5.78206540293048254433526659614, −5.26836639765476518793701593507, −5.15856104475478820670343436765, −4.96258133061725132884736292131, −4.40480750608830820804195378064, −4.34810298678029247321739979694, −3.98099867076858959594112509507, −3.40824580271926980109637073808, −3.32374310164769577599957698701, −2.98591441177090910463000932703, −2.62364377766526440657683146098, −2.36942790248762928598069209419, −2.03574209211811604778092805490, −1.46767376751878715206919523584, −1.40164450862598063698095984978, −0.954202248102399292379190874149, 0, 0, 0,
0.954202248102399292379190874149, 1.40164450862598063698095984978, 1.46767376751878715206919523584, 2.03574209211811604778092805490, 2.36942790248762928598069209419, 2.62364377766526440657683146098, 2.98591441177090910463000932703, 3.32374310164769577599957698701, 3.40824580271926980109637073808, 3.98099867076858959594112509507, 4.34810298678029247321739979694, 4.40480750608830820804195378064, 4.96258133061725132884736292131, 5.15856104475478820670343436765, 5.26836639765476518793701593507, 5.78206540293048254433526659614, 5.87110118822486464694552313330, 6.21041422458127661637363621755, 6.31583635588837666740889752562, 6.92068287342766729979839919191, 6.93869053722156150634991688501, 7.32621853969158218206902649164, 7.39613262024394233978612992189, 7.51920276436931709981823479390, 8.030114330952035488021400952348
