| L(s) = 1 | − 2·2-s − 12·3-s − 3·4-s + 6·5-s + 24·6-s + 14·7-s − 4·8-s + 90·9-s − 12·10-s − 40·11-s + 36·12-s − 28·14-s − 72·15-s + 15·16-s + 98·17-s − 180·18-s − 124·19-s − 18·20-s − 168·21-s + 80·22-s + 104·23-s + 48·24-s − 261·25-s − 540·27-s − 42·28-s + 194·29-s + 144·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s − 3/8·4-s + 0.536·5-s + 1.63·6-s + 0.755·7-s − 0.176·8-s + 10/3·9-s − 0.379·10-s − 1.09·11-s + 0.866·12-s − 0.534·14-s − 1.23·15-s + 0.234·16-s + 1.39·17-s − 2.35·18-s − 1.49·19-s − 0.201·20-s − 1.74·21-s + 0.775·22-s + 0.942·23-s + 0.408·24-s − 2.08·25-s − 3.84·27-s − 0.283·28-s + 1.24·29-s + 0.876·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2867461778\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2867461778\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + p T + 7 T^{2} + 3 p^{3} T^{3} + 31 p T^{4} + 3 p^{6} T^{5} + 7 p^{6} T^{6} + p^{10} T^{7} + p^{12} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 - 6 T + 297 T^{2} - 2094 T^{3} + 49084 T^{4} - 2094 p^{3} T^{5} + 297 p^{6} T^{6} - 6 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 2 p T + 249 T^{2} - 1754 T^{3} + 141728 T^{4} - 1754 p^{3} T^{5} + 249 p^{6} T^{6} - 2 p^{10} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 40 T + 4068 T^{2} + 116616 T^{3} + 7313302 T^{4} + 116616 p^{3} T^{5} + 4068 p^{6} T^{6} + 40 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 98 T + 12397 T^{2} - 986190 T^{3} + 96109736 T^{4} - 986190 p^{3} T^{5} + 12397 p^{6} T^{6} - 98 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 124 T + 21504 T^{2} + 1798684 T^{3} + 202620494 T^{4} + 1798684 p^{3} T^{5} + 21504 p^{6} T^{6} + 124 p^{9} T^{7} + p^{12} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 104 T + 41844 T^{2} - 3340584 T^{3} + 719588614 T^{4} - 3340584 p^{3} T^{5} + 41844 p^{6} T^{6} - 104 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 194 T + 64761 T^{2} - 7214754 T^{3} + 1694674348 T^{4} - 7214754 p^{3} T^{5} + 64761 p^{6} T^{6} - 194 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 26 T + 38189 T^{2} - 5395494 T^{3} + 828557428 T^{4} - 5395494 p^{3} T^{5} + 38189 p^{6} T^{6} + 26 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 102 T + 193281 T^{2} + 14822902 T^{3} + 14476248876 T^{4} + 14822902 p^{3} T^{5} + 193281 p^{6} T^{6} + 102 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 1054 T + 633469 T^{2} - 261177138 T^{3} + 78839658968 T^{4} - 261177138 p^{3} T^{5} + 633469 p^{6} T^{6} - 1054 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 450 T + 276753 T^{2} - 75560470 T^{3} + 29002070616 T^{4} - 75560470 p^{3} T^{5} + 276753 p^{6} T^{6} - 450 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 96 T - 19308 T^{2} - 124896 T^{3} + 17303792422 T^{4} - 124896 p^{3} T^{5} - 19308 p^{6} T^{6} - 96 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 262 T + 483789 T^{2} - 119532570 T^{3} + 100466115904 T^{4} - 119532570 p^{3} T^{5} + 483789 p^{6} T^{6} - 262 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 308 T + 557680 T^{2} + 98956308 T^{3} + 142593856142 T^{4} + 98956308 p^{3} T^{5} + 557680 p^{6} T^{6} + 308 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 928 T + 1041074 T^{2} + 583837824 T^{3} + 364336742755 T^{4} + 583837824 p^{3} T^{5} + 1041074 p^{6} T^{6} + 928 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1134 T + 1218801 T^{2} - 918506522 T^{3} + 562458950376 T^{4} - 918506522 p^{3} T^{5} + 1218801 p^{6} T^{6} - 1134 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 1064 T + 1201044 T^{2} + 690200040 T^{3} + 504474394630 T^{4} + 690200040 p^{3} T^{5} + 1201044 p^{6} T^{6} + 1064 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 952 T + 768042 T^{2} + 3278032 p T^{3} + 174760946603 T^{4} + 3278032 p^{4} T^{5} + 768042 p^{6} T^{6} + 952 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 746 T + 2156493 T^{2} + 1122200666 T^{3} + 1640976331028 T^{4} + 1122200666 p^{3} T^{5} + 2156493 p^{6} T^{6} + 746 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 404 T + 21824 p T^{2} - 594579060 T^{3} + 1476544322126 T^{4} - 594579060 p^{3} T^{5} + 21824 p^{7} T^{6} - 404 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1620 T + 2005896 T^{2} + 1665077532 T^{3} + 1565610279790 T^{4} + 1665077532 p^{3} T^{5} + 2005896 p^{6} T^{6} + 1620 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2166 T + 4160745 T^{2} + 5948172454 T^{3} + 5928410141748 T^{4} + 5948172454 p^{3} T^{5} + 4160745 p^{6} T^{6} + 2166 p^{9} T^{7} + p^{12} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.43556213371228044364761210493, −7.23929082655130372529532620872, −7.03992908244070721783683507215, −6.62074205454994295777568359573, −6.37374558639861122191615688450, −6.03815203136158766908090607279, −5.78805966745361168448709319353, −5.73363883970747809119022688912, −5.69408848588139688180208535462, −5.46299961969421959997042950031, −4.75199752650937116536138547260, −4.72543190003344406896896238866, −4.70461504707280768006222646963, −4.14413887398902340422365592836, −3.98940217673127244186006041620, −3.83020799620250816053407238478, −2.99122005000924061146634408929, −2.81803957325702872135780170184, −2.59128616376641499464584152703, −1.86981230770441601724898593627, −1.84128137845546088408158905928, −1.29046441567326641804999939770, −0.74850480592895428022114320089, −0.71098487746157144149232049275, −0.16157998886395888825818635129,
0.16157998886395888825818635129, 0.71098487746157144149232049275, 0.74850480592895428022114320089, 1.29046441567326641804999939770, 1.84128137845546088408158905928, 1.86981230770441601724898593627, 2.59128616376641499464584152703, 2.81803957325702872135780170184, 2.99122005000924061146634408929, 3.83020799620250816053407238478, 3.98940217673127244186006041620, 4.14413887398902340422365592836, 4.70461504707280768006222646963, 4.72543190003344406896896238866, 4.75199752650937116536138547260, 5.46299961969421959997042950031, 5.69408848588139688180208535462, 5.73363883970747809119022688912, 5.78805966745361168448709319353, 6.03815203136158766908090607279, 6.37374558639861122191615688450, 6.62074205454994295777568359573, 7.03992908244070721783683507215, 7.23929082655130372529532620872, 7.43556213371228044364761210493
