| L(s) = 1 | − 4·2-s + 2·3-s + 10·4-s − 8·6-s + 2·7-s − 20·8-s − 7·9-s + 20·12-s − 6·13-s − 8·14-s + 35·16-s − 2·17-s + 28·18-s − 6·19-s + 4·21-s − 40·24-s + 24·26-s − 20·27-s + 20·28-s − 14·29-s − 16·31-s − 56·32-s + 8·34-s − 70·36-s + 16·37-s + 24·38-s − 12·39-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.15·3-s + 5·4-s − 3.26·6-s + 0.755·7-s − 7.07·8-s − 7/3·9-s + 5.77·12-s − 1.66·13-s − 2.13·14-s + 35/4·16-s − 0.485·17-s + 6.59·18-s − 1.37·19-s + 0.872·21-s − 8.16·24-s + 4.70·26-s − 3.84·27-s + 3.77·28-s − 2.59·29-s − 2.87·31-s − 9.89·32-s + 1.37·34-s − 11.6·36-s + 2.63·37-s + 3.89·38-s − 1.92·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.367468539\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.367468539\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 4 T^{2} + 18 T^{3} - 22 T^{4} + 18 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 46 T^{2} + 194 T^{3} + 878 T^{4} + 194 p T^{5} + 46 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 39 T^{2} + 22 T^{3} + 753 T^{4} + 22 p T^{5} + 39 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 59 T^{2} + 314 T^{3} + 1569 T^{4} + 314 p T^{5} + 59 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 54 T^{2} + 100 T^{3} + 1382 T^{4} + 100 p T^{5} + 54 p^{2} T^{6} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 + 14 T + 150 T^{2} + 1098 T^{3} + 6638 T^{4} + 1098 p T^{5} + 150 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 16 T + 158 T^{2} + 1028 T^{3} + 6094 T^{4} + 1028 p T^{5} + 158 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 206 T^{2} - 44 p T^{3} + 11710 T^{4} - 44 p^{2} T^{5} + 206 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 111 T^{2} + 900 T^{3} + 5861 T^{4} + 900 p T^{5} + 111 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 5 T + 91 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 16 T + 222 T^{2} - 1796 T^{3} + 14670 T^{4} - 1796 p T^{5} + 222 p^{2} T^{6} - 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 22 T + 368 T^{2} - 3918 T^{3} + 33770 T^{4} - 3918 p T^{5} + 368 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 243 T^{2} - 1630 T^{3} + 21473 T^{4} - 1630 p T^{5} + 243 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 194 T^{2} + 1430 T^{3} + 17326 T^{4} + 1430 p T^{5} + 194 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 - 10 T + 171 T^{2} - 800 T^{3} + 11637 T^{4} - 800 p T^{5} + 171 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 - 26 T + 448 T^{2} - 5038 T^{3} + 48154 T^{4} - 5038 p T^{5} + 448 p^{2} T^{6} - 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 275 T^{2} - 834 T^{3} + 29541 T^{4} - 834 p T^{5} + 275 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 174 T^{2} - 1854 T^{3} + 186 p T^{4} - 1854 p T^{5} + 174 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 - 18 T + 383 T^{2} - 4192 T^{3} + 49745 T^{4} - 4192 p T^{5} + 383 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - T + 147 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 28 T + 527 T^{2} - 6650 T^{3} + 73201 T^{4} - 6650 p T^{5} + 527 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−5.90640618423987594758079971311, −5.52064973081375210050433117022, −5.36382015500734482450647932112, −5.35595385755993560884309928177, −5.18020455501476165798899117442, −4.57904097062063562202206564453, −4.53882225070578234987588108284, −4.40220981444924775835342372351, −3.99012914720900419912512955460, −3.63516046686881548145295325255, −3.50366904211931937331949914434, −3.43670395603823572306270757825, −3.43505249007845580105435325179, −2.80708778505931958293958894684, −2.64842136853522845420564049785, −2.43672120931983380350054118784, −2.28163105833482027908763529609, −2.18530131942206774039908611011, −1.98381435689543591747529791337, −1.81203503120474547994112144264, −1.80318562821606905046968473732, −0.809326013302770767661986270804, −0.61628663680237980528323594755, −0.55749106933724673210009596108, −0.40993270281253365088610506060,
0.40993270281253365088610506060, 0.55749106933724673210009596108, 0.61628663680237980528323594755, 0.809326013302770767661986270804, 1.80318562821606905046968473732, 1.81203503120474547994112144264, 1.98381435689543591747529791337, 2.18530131942206774039908611011, 2.28163105833482027908763529609, 2.43672120931983380350054118784, 2.64842136853522845420564049785, 2.80708778505931958293958894684, 3.43505249007845580105435325179, 3.43670395603823572306270757825, 3.50366904211931937331949914434, 3.63516046686881548145295325255, 3.99012914720900419912512955460, 4.40220981444924775835342372351, 4.53882225070578234987588108284, 4.57904097062063562202206564453, 5.18020455501476165798899117442, 5.35595385755993560884309928177, 5.36382015500734482450647932112, 5.52064973081375210050433117022, 5.90640618423987594758079971311
