L(s) = 1 | − 4·2-s + 2·3-s + 10·4-s − 2·5-s − 8·6-s − 3·7-s − 20·8-s + 4·9-s + 8·10-s + 20·12-s − 13-s + 12·14-s − 4·15-s + 35·16-s + 7·17-s − 16·18-s − 4·19-s − 20·20-s − 6·21-s − 5·23-s − 40·24-s − 4·25-s + 4·26-s + 6·27-s − 30·28-s + 14·29-s + 16·30-s + ⋯ |
L(s) = 1 | − 2.82·2-s + 1.15·3-s + 5·4-s − 0.894·5-s − 3.26·6-s − 1.13·7-s − 7.07·8-s + 4/3·9-s + 2.52·10-s + 5.77·12-s − 0.277·13-s + 3.20·14-s − 1.03·15-s + 35/4·16-s + 1.69·17-s − 3.77·18-s − 0.917·19-s − 4.47·20-s − 1.30·21-s − 1.04·23-s − 8.16·24-s − 4/5·25-s + 0.784·26-s + 1.15·27-s − 5.66·28-s + 2.59·29-s + 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3756683676\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3756683676\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 2 | $C_1$ | \( ( 1 + T )^{4} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
good | 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 T^{3} - T^{4} + 2 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 2 T + 8 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 3 T + p T^{2} - 4 T^{3} - 48 T^{4} - 4 p T^{5} + p^{3} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + T + 22 T^{2} + 59 T^{3} + 2 p^{2} T^{4} + 59 p T^{5} + 22 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 7 T + 62 T^{2} - 261 T^{3} + 1398 T^{4} - 261 p T^{5} + 62 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 5 T + 71 T^{2} + 312 T^{3} + 2304 T^{4} + 312 p T^{5} + 71 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 14 T + 146 T^{2} - 1134 T^{3} + 6717 T^{4} - 1134 p T^{5} + 146 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 4 T + 76 T^{2} - 212 T^{3} + 3062 T^{4} - 212 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 12 T + 70 T^{2} + 118 T^{3} - 2127 T^{4} + 118 p T^{5} + 70 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 12 T + 146 T^{2} + 966 T^{3} + 7638 T^{4} + 966 p T^{5} + 146 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 14 T + 52 T^{2} - 34 T^{3} - 14 p T^{4} - 34 p T^{5} + 52 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 206 T^{2} + 1524 T^{3} + 14595 T^{4} + 1524 p T^{5} + 206 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 10 T + 206 T^{2} + 1386 T^{3} + 16005 T^{4} + 1386 p T^{5} + 206 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 3 T + 191 T^{2} - 378 T^{3} + 15684 T^{4} - 378 p T^{5} + 191 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 6 T + 112 T^{2} + 872 T^{3} + 8634 T^{4} + 872 p T^{5} + 112 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 15 T + 184 T^{2} - 887 T^{3} + 7806 T^{4} - 887 p T^{5} + 184 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 16 T + 230 T^{2} + 1950 T^{3} + 19662 T^{4} + 1950 p T^{5} + 230 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 9 T + 298 T^{2} - 1939 T^{3} + 32814 T^{4} - 1939 p T^{5} + 298 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 20 T + 394 T^{2} + 4706 T^{3} + 49678 T^{4} + 4706 p T^{5} + 394 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 22 T + 500 T^{2} - 5988 T^{3} + 69714 T^{4} - 5988 p T^{5} + 500 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 278 T^{2} + 1854 T^{3} + 35094 T^{4} + 1854 p T^{5} + 278 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−6.06544363201791438445262211159, −5.93598411124796327461434089839, −5.50383847510715975467682487653, −5.49644774318388026950052378899, −5.13916584474663823300265323969, −4.67199037979901755125478974121, −4.60260322193221114252244589636, −4.57769294656834178492125926709, −4.25785499814532684921989671350, −3.95354267075113859127260151226, −3.60019744387762076624548847487, −3.53581338379209747134411225842, −3.18783009424096015679398584660, −3.08132740616861052546796526076, −3.05119658116346577158174961752, −2.86070650430628452624297574564, −2.44280148020333503669290156801, −2.04416015836813858265891006576, −1.97280050037077129562492170259, −1.86396118842654118983588011147, −1.42486077276403594445916255565, −1.16892330791519608977018898783, −0.887361480785812721843903231440, −0.51610211686086639244445759792, −0.18003325381719503176537516332,
0.18003325381719503176537516332, 0.51610211686086639244445759792, 0.887361480785812721843903231440, 1.16892330791519608977018898783, 1.42486077276403594445916255565, 1.86396118842654118983588011147, 1.97280050037077129562492170259, 2.04416015836813858265891006576, 2.44280148020333503669290156801, 2.86070650430628452624297574564, 3.05119658116346577158174961752, 3.08132740616861052546796526076, 3.18783009424096015679398584660, 3.53581338379209747134411225842, 3.60019744387762076624548847487, 3.95354267075113859127260151226, 4.25785499814532684921989671350, 4.57769294656834178492125926709, 4.60260322193221114252244589636, 4.67199037979901755125478974121, 5.13916584474663823300265323969, 5.49644774318388026950052378899, 5.50383847510715975467682487653, 5.93598411124796327461434089839, 6.06544363201791438445262211159