| L(s) = 1 | − 2-s − 4·3-s + 4·6-s − 4·7-s + 10·9-s − 2·11-s − 4·13-s + 4·14-s − 16-s + 2·17-s − 10·18-s + 7·19-s + 16·21-s + 2·22-s − 3·23-s + 4·26-s − 20·27-s + 29-s + 3·31-s + 32-s + 8·33-s − 2·34-s − 10·37-s − 7·38-s + 16·39-s − 16·41-s − 16·42-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 2.30·3-s + 1.63·6-s − 1.51·7-s + 10/3·9-s − 0.603·11-s − 1.10·13-s + 1.06·14-s − 1/4·16-s + 0.485·17-s − 2.35·18-s + 1.60·19-s + 3.49·21-s + 0.426·22-s − 0.625·23-s + 0.784·26-s − 3.84·27-s + 0.185·29-s + 0.538·31-s + 0.176·32-s + 1.39·33-s − 0.342·34-s − 1.64·37-s − 1.13·38-s + 2.56·39-s − 2.49·41-s − 2.46·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + T^{2} + T^{3} + p T^{4} + p T^{5} + p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 20 T^{2} + 34 T^{3} + 294 T^{4} + 34 p T^{5} + 20 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 2 T + 40 T^{2} - 62 T^{3} + 878 T^{4} - 62 p T^{5} + 40 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 7 T + 64 T^{2} - 351 T^{3} + 1774 T^{4} - 351 p T^{5} + 64 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 3 T + 40 T^{2} - 49 T^{3} + 494 T^{4} - 49 p T^{5} + 40 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 86 T^{2} - 35 T^{3} + 3378 T^{4} - 35 p T^{5} + 86 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 3 T - 4 T^{2} - 119 T^{3} + 58 p T^{4} - 119 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 10 T + 64 T^{2} + 270 T^{3} + 1870 T^{4} + 270 p T^{5} + 64 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 16 T + 4 p T^{2} + 1280 T^{3} + 8694 T^{4} + 1280 p T^{5} + 4 p^{3} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 3 T + 128 T^{2} + 275 T^{3} + 7246 T^{4} + 275 p T^{5} + 128 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 5 T + 148 T^{2} + 689 T^{3} + 9638 T^{4} + 689 p T^{5} + 148 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 5 T + 174 T^{2} + 727 T^{3} + 12802 T^{4} + 727 p T^{5} + 174 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 20 T + 316 T^{2} + 3236 T^{3} + 28790 T^{4} + 3236 p T^{5} + 316 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 12 T + 180 T^{2} - 1508 T^{3} + 15014 T^{4} - 1508 p T^{5} + 180 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 22 T + 228 T^{2} - 1254 T^{3} + 6086 T^{4} - 1254 p T^{5} + 228 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 52 T^{2} + 304 T^{3} + 7478 T^{4} + 304 p T^{5} + 52 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 13 T + 126 T^{2} + 261 T^{3} - 3934 T^{4} + 261 p T^{5} + 126 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 11 T + 196 T^{2} - 1167 T^{3} + 15030 T^{4} - 1167 p T^{5} + 196 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + T + 296 T^{2} + 329 T^{3} + 35310 T^{4} + 329 p T^{5} + 296 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 5 T + 194 T^{2} + 139 T^{3} + 16986 T^{4} + 139 p T^{5} + 194 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 17 T + 374 T^{2} - 4127 T^{3} + 52210 T^{4} - 4127 p T^{5} + 374 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−6.03429821038264679000110460504, −5.63971272897136047611339106591, −5.45419471882198326642552180419, −5.38017444008633640856405664451, −5.35595716707795955498554441164, −5.17777409086307531692665007848, −4.94316040221957707449605244437, −4.90602739294576430714207333614, −4.57560709740909190022191259761, −4.27819732416698600139942581816, −4.11275103404571954533667826650, −3.93905012722121507356486663713, −3.70262640882965190000898562839, −3.38799020803252584301652601120, −3.27082962339334669522211147717, −3.24742665149066906072433469412, −2.79621608301281199676603820199, −2.53732933501111438252453360909, −2.52243183091789510497963349654, −1.99176659221867831715904578705, −1.80036371757981088065667632756, −1.66967461128626140280419849002, −1.10019859197329639423926110918, −1.03504640441414734617922988468, −0.906226939728232026591100592178, 0, 0, 0, 0,
0.906226939728232026591100592178, 1.03504640441414734617922988468, 1.10019859197329639423926110918, 1.66967461128626140280419849002, 1.80036371757981088065667632756, 1.99176659221867831715904578705, 2.52243183091789510497963349654, 2.53732933501111438252453360909, 2.79621608301281199676603820199, 3.24742665149066906072433469412, 3.27082962339334669522211147717, 3.38799020803252584301652601120, 3.70262640882965190000898562839, 3.93905012722121507356486663713, 4.11275103404571954533667826650, 4.27819732416698600139942581816, 4.57560709740909190022191259761, 4.90602739294576430714207333614, 4.94316040221957707449605244437, 5.17777409086307531692665007848, 5.35595716707795955498554441164, 5.38017444008633640856405664451, 5.45419471882198326642552180419, 5.63971272897136047611339106591, 6.03429821038264679000110460504
