| L(s) = 1 | − 2-s + 2·4-s − 4·5-s + 7-s − 8-s + 4·10-s − 4·11-s + 7·13-s − 14-s + 5·17-s + 9·19-s − 8·20-s + 4·22-s − 14·23-s + 5·25-s − 7·26-s + 2·28-s + 6·29-s − 2·31-s + 8·32-s − 5·34-s − 4·35-s + 3·37-s − 9·38-s + 4·40-s + 2·41-s − 21·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 1.78·5-s + 0.377·7-s − 0.353·8-s + 1.26·10-s − 1.20·11-s + 1.94·13-s − 0.267·14-s + 1.21·17-s + 2.06·19-s − 1.78·20-s + 0.852·22-s − 2.91·23-s + 25-s − 1.37·26-s + 0.377·28-s + 1.11·29-s − 0.359·31-s + 1.41·32-s − 0.857·34-s − 0.676·35-s + 0.493·37-s − 1.45·38-s + 0.632·40-s + 0.312·41-s − 3.20·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.708370919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.708370919\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $S_4\times C_2$ | \( 1 + T - T^{2} - p T^{3} + T^{4} - p^{2} T^{5} - p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 4 T + 11 T^{2} + 31 T^{3} + 91 T^{4} + 31 p T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - T + 22 T^{2} - 16 T^{3} + 214 T^{4} - 16 p T^{5} + 22 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 7 T + 37 T^{2} - 118 T^{3} + 466 T^{4} - 118 p T^{5} + 37 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 5 T + 44 T^{2} - 86 T^{3} + 682 T^{4} - 86 p T^{5} + 44 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 9 T + 4 p T^{2} - 432 T^{3} + 2112 T^{4} - 432 p T^{5} + 4 p^{3} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 14 T + 143 T^{2} + 953 T^{3} + 5332 T^{4} + 953 p T^{5} + 143 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 6 T + 107 T^{2} - 417 T^{3} + 4374 T^{4} - 417 p T^{5} + 107 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 2 T + 103 T^{2} + 113 T^{3} + 4405 T^{4} + 113 p T^{5} + 103 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 3 T + 67 T^{2} - 189 T^{3} + 2277 T^{4} - 189 p T^{5} + 67 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 2 T + 101 T^{2} - 407 T^{3} + 4894 T^{4} - 407 p T^{5} + 101 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 21 T + 328 T^{2} + 3186 T^{3} + 25008 T^{4} + 3186 p T^{5} + 328 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 7 T + 173 T^{2} - 985 T^{3} + 11845 T^{4} - 985 p T^{5} + 173 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 6 T + 167 T^{2} - 789 T^{3} + 11961 T^{4} - 789 p T^{5} + 167 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 2 T + 215 T^{2} + 305 T^{3} + 18421 T^{4} + 305 p T^{5} + 215 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 15 T + 322 T^{2} - 2910 T^{3} + 31962 T^{4} - 2910 p T^{5} + 322 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 14 T + 247 T^{2} - 1979 T^{3} + 21949 T^{4} - 1979 p T^{5} + 247 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 3 T + 197 T^{2} - 909 T^{3} + 17697 T^{4} - 909 p T^{5} + 197 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 22 T + 421 T^{2} - 4807 T^{3} + 49780 T^{4} - 4807 p T^{5} + 421 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 11 T + 292 T^{2} - 2582 T^{3} + 33694 T^{4} - 2582 p T^{5} + 292 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 18 T + 329 T^{2} + 3771 T^{3} + 42384 T^{4} + 3771 p T^{5} + 329 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 6 T + 224 T^{2} - 1554 T^{3} + 26094 T^{4} - 1554 p T^{5} + 224 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 26 T + 544 T^{2} - 7460 T^{3} + 85489 T^{4} - 7460 p T^{5} + 544 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−7.33159578143784533169593356503, −7.00012159390854052698004566836, −6.99515174042409879718721289538, −6.62825812796130266272600892064, −6.30935408181637199032909397485, −6.29923910325706021828973659016, −5.96849902777359052968277194702, −5.59505433753931479950671509318, −5.50368069580801506486047040498, −5.19308608666695294416251919483, −4.87142897534026215696996313648, −4.64973936865059218636440814893, −4.54437380029603670247969439808, −3.86424979649266161580674270239, −3.78772132931444710280312675983, −3.68461309018722344999261665210, −3.31236515339559816667298231151, −3.20428264604883994995032021875, −2.85960212887292866761490665200, −2.31565266577776637577914817395, −1.92872229089309196452290458560, −1.91226122146110316950392342667, −1.18617839694837518367140819035, −0.892194307395982919396068311152, −0.45448446258251434250283040992,
0.45448446258251434250283040992, 0.892194307395982919396068311152, 1.18617839694837518367140819035, 1.91226122146110316950392342667, 1.92872229089309196452290458560, 2.31565266577776637577914817395, 2.85960212887292866761490665200, 3.20428264604883994995032021875, 3.31236515339559816667298231151, 3.68461309018722344999261665210, 3.78772132931444710280312675983, 3.86424979649266161580674270239, 4.54437380029603670247969439808, 4.64973936865059218636440814893, 4.87142897534026215696996313648, 5.19308608666695294416251919483, 5.50368069580801506486047040498, 5.59505433753931479950671509318, 5.96849902777359052968277194702, 6.29923910325706021828973659016, 6.30935408181637199032909397485, 6.62825812796130266272600892064, 6.99515174042409879718721289538, 7.00012159390854052698004566836, 7.33159578143784533169593356503
