| L(s) = 1 | + 3-s + 2·5-s + 2·9-s + 11-s − 2·13-s + 2·15-s + 11·17-s − 6·19-s + 2·23-s + 25-s − 27-s + 10·29-s − 4·31-s + 33-s − 2·39-s − 12·41-s − 12·43-s + 4·45-s + 9·47-s + 11·51-s + 18·53-s + 2·55-s − 6·57-s − 8·59-s − 22·61-s − 4·65-s + 12·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 2/3·9-s + 0.301·11-s − 0.554·13-s + 0.516·15-s + 2.66·17-s − 1.37·19-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 1.85·29-s − 0.718·31-s + 0.174·33-s − 0.320·39-s − 1.87·41-s − 1.82·43-s + 0.596·45-s + 1.31·47-s + 1.54·51-s + 2.47·53-s + 0.269·55-s − 0.794·57-s − 1.04·59-s − 2.81·61-s − 0.496·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.562005719\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.562005719\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - T - T^{2} + 4 T^{3} - 8 T^{4} + 4 p T^{5} - p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - T - 17 T^{2} + 4 T^{3} + 192 T^{4} + 4 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 11 T + 61 T^{2} - 286 T^{3} + 1254 T^{4} - 286 p T^{5} + 61 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 6 T + 6 T^{2} - 48 T^{3} - 145 T^{4} - 48 p T^{5} + 6 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 2 T - 26 T^{2} + 32 T^{3} + 279 T^{4} + 32 p T^{5} - 26 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 5 T + 60 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 4 T + 18 T^{2} - 256 T^{3} - 1453 T^{4} - 256 p T^{5} + 18 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 6 T^{2} - 1333 T^{4} - 6 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 6 T + 74 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 9 T - 29 T^{2} - 144 T^{3} + 6084 T^{4} - 144 p T^{5} - 29 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 18 T + 154 T^{2} - 1152 T^{3} + 8919 T^{4} - 1152 p T^{5} + 154 p^{2} T^{6} - 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 + 22 T + 258 T^{2} + 2288 T^{3} + 17831 T^{4} + 2288 p T^{5} + 258 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 12 T + 42 T^{2} + 384 T^{3} - 3733 T^{4} + 384 p T^{5} + 42 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $D_4\times C_2$ | \( 1 - 8 T - 30 T^{2} + 416 T^{3} - 1165 T^{4} + 416 p T^{5} - 30 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 11 T - 29 T^{2} + 88 T^{3} + 6700 T^{4} + 88 p T^{5} - 29 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{4} \) |
| 89 | $D_4\times C_2$ | \( 1 - 2 T - 158 T^{2} + 32 T^{3} + 17967 T^{4} + 32 p T^{5} - 158 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 15 T + 212 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 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
−6.77329055118000203873575899924, −6.09495180027569976670453402781, −6.05294578203701796113130799455, −5.98030808831399409839303244947, −5.67848868732071471174292024960, −5.59044226854599545434220826296, −5.18165680818807270454175604096, −5.13472693775090626074507189276, −4.79214651036639063929924069982, −4.49941419469602688282933060173, −4.49345930144745702437824573195, −4.21266694781622261577274645182, −3.74362940032869252125142011956, −3.53549871345603245393771640207, −3.52421233915913398641555951370, −3.16831753622017040497641810319, −2.91945731425704302135340834025, −2.67882262194379045017227871740, −2.28211865435842905960583466534, −2.17273841141671272030388473004, −1.83930142815929609664240333550, −1.35491559699654480310465303788, −1.26732189887941605994062395590, −1.05184547559119673258041889096, −0.23164004417153024271393660749,
0.23164004417153024271393660749, 1.05184547559119673258041889096, 1.26732189887941605994062395590, 1.35491559699654480310465303788, 1.83930142815929609664240333550, 2.17273841141671272030388473004, 2.28211865435842905960583466534, 2.67882262194379045017227871740, 2.91945731425704302135340834025, 3.16831753622017040497641810319, 3.52421233915913398641555951370, 3.53549871345603245393771640207, 3.74362940032869252125142011956, 4.21266694781622261577274645182, 4.49345930144745702437824573195, 4.49941419469602688282933060173, 4.79214651036639063929924069982, 5.13472693775090626074507189276, 5.18165680818807270454175604096, 5.59044226854599545434220826296, 5.67848868732071471174292024960, 5.98030808831399409839303244947, 6.05294578203701796113130799455, 6.09495180027569976670453402781, 6.77329055118000203873575899924
