| 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
| \(L(1)\) |
\(\approx\) |
\(13.17522840\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.17522840\) |
| \(L(\frac{3}{2})\) |
|
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
−7.03918028733771430002244040826, −6.92663373165416272664010882377, −6.56094297548749883915109378461, −6.52184127345309824142064906821, −6.43592890155355658604529681843, −6.34637654179161476578322413788, −5.85091591953721541984394674515, −5.59161820351733683341396949645, −5.44989752820091883251906654476, −5.08088511320187006261912761698, −4.93394312574895959686317806507, −4.91428674961683872377172788555, −4.72846956073830066608982671363, −3.84890414268593156093923316644, −3.81486043970888463737891111458, −3.66654777076410773472041825319, −3.23002603768436519332108956233, −3.20958398365823596806459979606, −2.93414236632483147791398910084, −2.22023280754702886723506608444, −2.12805383611502224440517804319, −1.71275901497551536766036693485, −1.59029122202617954127435993087, −1.22497281734807156992028915967, −0.73433714241970085875408554675,
0.73433714241970085875408554675, 1.22497281734807156992028915967, 1.59029122202617954127435993087, 1.71275901497551536766036693485, 2.12805383611502224440517804319, 2.22023280754702886723506608444, 2.93414236632483147791398910084, 3.20958398365823596806459979606, 3.23002603768436519332108956233, 3.66654777076410773472041825319, 3.81486043970888463737891111458, 3.84890414268593156093923316644, 4.72846956073830066608982671363, 4.91428674961683872377172788555, 4.93394312574895959686317806507, 5.08088511320187006261912761698, 5.44989752820091883251906654476, 5.59161820351733683341396949645, 5.85091591953721541984394674515, 6.34637654179161476578322413788, 6.43592890155355658604529681843, 6.52184127345309824142064906821, 6.56094297548749883915109378461, 6.92663373165416272664010882377, 7.03918028733771430002244040826
