| L(s) = 1 | + 2-s + 3-s + 2·4-s + 7·5-s + 6-s − 7-s + 7·10-s + 11-s + 2·12-s + 4·13-s − 14-s + 7·15-s − 12·17-s + 19-s + 14·20-s − 21-s + 22-s + 10·23-s + 20·25-s + 4·26-s − 2·28-s + 2·29-s + 7·30-s + 17·31-s − 11·32-s + 33-s − 12·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 4-s + 3.13·5-s + 0.408·6-s − 0.377·7-s + 2.21·10-s + 0.301·11-s + 0.577·12-s + 1.10·13-s − 0.267·14-s + 1.80·15-s − 2.91·17-s + 0.229·19-s + 3.13·20-s − 0.218·21-s + 0.213·22-s + 2.08·23-s + 4·25-s + 0.784·26-s − 0.377·28-s + 0.371·29-s + 1.27·30-s + 3.05·31-s − 1.94·32-s + 0.174·33-s − 2.05·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 7^{4} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.082602107\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.082602107\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_4$ | \( 1 - T + T^{2} - T^{3} + T^{4} \) |
| 7 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 11 | $C_4$ | \( 1 - T - 9 T^{2} - p T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T - T^{2} + 3 T^{3} - T^{4} + 3 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 7 T + 29 T^{2} - 93 T^{3} + 236 T^{4} - 93 p T^{5} + 29 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 4 T + 3 T^{2} - 50 T^{3} + 341 T^{4} - 50 p T^{5} + 3 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 180 p T^{5} + 37 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - T - 13 T^{2} - 53 T^{3} + 400 T^{4} - 53 p T^{5} - 13 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 5 T + 21 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 2 T - 25 T^{2} + 108 T^{3} + 509 T^{4} + 108 p T^{5} - 25 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 17 T + 153 T^{2} - 1129 T^{3} + 7100 T^{4} - 1129 p T^{5} + 153 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 6 T + 99 T^{2} + 272 T^{3} + 3969 T^{4} + 272 p T^{5} + 99 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 49 T^{2} - 60 T^{3} + 1861 T^{4} - 60 p T^{5} + 49 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 2 T + 82 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_4\times C_2$ | \( 1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 180 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 26 T + 323 T^{2} + 2870 T^{3} + 22001 T^{4} + 2870 p T^{5} + 323 p^{2} T^{6} + 26 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 14 T + 37 T^{2} - 758 T^{3} - 10095 T^{4} - 758 p T^{5} + 37 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_4\times C_2$ | \( 1 + 20 T + 179 T^{2} + 1600 T^{3} + 15001 T^{4} + 1600 p T^{5} + 179 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{4} \) |
| 71 | $C_4\times C_2$ | \( 1 + 20 T + 169 T^{2} + 1600 T^{3} + 17121 T^{4} + 1600 p T^{5} + 169 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 2 T - 49 T^{2} + 406 T^{3} + 5929 T^{4} + 406 p T^{5} - 49 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 - 14 T + 117 T^{2} - 532 T^{3} - 1795 T^{4} - 532 p T^{5} + 117 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 14 T + 193 T^{2} + 2140 T^{3} + 26421 T^{4} + 2140 p T^{5} + 193 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 9 T + 167 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 2 T + 27 T^{2} + 760 T^{3} + 10181 T^{4} + 760 p T^{5} + 27 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 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
−9.255310020287951855014939008779, −8.662948293132480471486146819153, −8.488663500706773981299758648074, −8.162266324129877762837448689735, −8.122857531790468164150808397081, −7.41801494672860857343269664993, −6.91017978197698429831544671721, −6.90213157301671675177539843498, −6.63935676954067093196231002282, −6.51695811239720675437272378095, −6.09573841132526209340137691233, −5.98646079563706530886480801015, −5.77584061055552413518218470610, −5.50262462335925024947368800969, −4.93479727158539279636699920102, −4.68856583053984370700299607531, −4.53877972703105823660575534005, −4.11901078430066018675588223709, −3.26339151780591006862708227484, −3.24015726984892818522877822382, −2.76004668971771154316473266580, −2.64889669748450863562112950920, −2.01714381392967332276919030370, −1.73056415830013496428354135776, −1.40865018718095934519670857234,
1.40865018718095934519670857234, 1.73056415830013496428354135776, 2.01714381392967332276919030370, 2.64889669748450863562112950920, 2.76004668971771154316473266580, 3.24015726984892818522877822382, 3.26339151780591006862708227484, 4.11901078430066018675588223709, 4.53877972703105823660575534005, 4.68856583053984370700299607531, 4.93479727158539279636699920102, 5.50262462335925024947368800969, 5.77584061055552413518218470610, 5.98646079563706530886480801015, 6.09573841132526209340137691233, 6.51695811239720675437272378095, 6.63935676954067093196231002282, 6.90213157301671675177539843498, 6.91017978197698429831544671721, 7.41801494672860857343269664993, 8.122857531790468164150808397081, 8.162266324129877762837448689735, 8.488663500706773981299758648074, 8.662948293132480471486146819153, 9.255310020287951855014939008779
