| L(s) = 1 | + 8·2-s + 4·3-s + 40·4-s + 32·6-s + 7-s + 160·8-s − 30·9-s − 39·11-s + 160·12-s + 20·13-s + 8·14-s + 560·16-s + 23·17-s − 240·18-s + 53·19-s + 4·21-s − 312·22-s + 92·23-s + 640·24-s + 160·26-s − 223·27-s + 40·28-s + 161·29-s + 388·31-s + 1.79e3·32-s − 156·33-s + 184·34-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 0.769·3-s + 5·4-s + 2.17·6-s + 0.0539·7-s + 7.07·8-s − 1.11·9-s − 1.06·11-s + 3.84·12-s + 0.426·13-s + 0.152·14-s + 35/4·16-s + 0.328·17-s − 3.14·18-s + 0.639·19-s + 0.0415·21-s − 3.02·22-s + 0.834·23-s + 5.44·24-s + 1.20·26-s − 1.58·27-s + 0.269·28-s + 1.03·29-s + 2.24·31-s + 9.89·32-s − 0.822·33-s + 0.928·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 5^{8} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(52.50790680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(52.50790680\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 - p T )^{4} \) |
| 5 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - p T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 4 T + 46 T^{2} - p^{4} T^{3} + 1190 T^{4} - p^{7} T^{5} + 46 p^{6} T^{6} - 4 p^{9} T^{7} + p^{12} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - T - 135 T^{2} - 521 p T^{3} + 125756 T^{4} - 521 p^{4} T^{5} - 135 p^{6} T^{6} - p^{9} T^{7} + p^{12} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 39 T + 323 p T^{2} + 61307 T^{3} + 4937428 T^{4} + 61307 p^{3} T^{5} + 323 p^{7} T^{6} + 39 p^{9} T^{7} + p^{12} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 20 T + 3840 T^{2} - 157301 T^{3} + 10278344 T^{4} - 157301 p^{3} T^{5} + 3840 p^{6} T^{6} - 20 p^{9} T^{7} + p^{12} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 23 T + 5447 T^{2} + 251497 T^{3} + 6295268 T^{4} + 251497 p^{3} T^{5} + 5447 p^{6} T^{6} - 23 p^{9} T^{7} + p^{12} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 53 T + 653 p T^{2} - 750501 T^{3} + 76029128 T^{4} - 750501 p^{3} T^{5} + 653 p^{7} T^{6} - 53 p^{9} T^{7} + p^{12} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 161 T + 2424 p T^{2} - 9893175 T^{3} + 2240848710 T^{4} - 9893175 p^{3} T^{5} + 2424 p^{7} T^{6} - 161 p^{9} T^{7} + p^{12} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 388 T + 132240 T^{2} - 29130015 T^{3} + 5707089166 T^{4} - 29130015 p^{3} T^{5} + 132240 p^{6} T^{6} - 388 p^{9} T^{7} + p^{12} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 466 T + 240532 T^{2} + 71494750 T^{3} + 19227960950 T^{4} + 71494750 p^{3} T^{5} + 240532 p^{6} T^{6} + 466 p^{9} T^{7} + p^{12} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 484 T + 239452 T^{2} - 82501249 T^{3} + 25012433496 T^{4} - 82501249 p^{3} T^{5} + 239452 p^{6} T^{6} - 484 p^{9} T^{7} + p^{12} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 894 T + 484780 T^{2} + 190414350 T^{3} + 59052057270 T^{4} + 190414350 p^{3} T^{5} + 484780 p^{6} T^{6} + 894 p^{9} T^{7} + p^{12} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 265 T + 184902 T^{2} - 39015297 T^{3} + 18141867922 T^{4} - 39015297 p^{3} T^{5} + 184902 p^{6} T^{6} - 265 p^{9} T^{7} + p^{12} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 576 T + 488156 T^{2} + 177483648 T^{3} + 97785187798 T^{4} + 177483648 p^{3} T^{5} + 488156 p^{6} T^{6} + 576 p^{9} T^{7} + p^{12} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 94 T + 578056 T^{2} + 37319902 T^{3} + 164753482782 T^{4} + 37319902 p^{3} T^{5} + 578056 p^{6} T^{6} + 94 p^{9} T^{7} + p^{12} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 1153 T + 853867 T^{2} - 469738549 T^{3} + 241810092932 T^{4} - 469738549 p^{3} T^{5} + 853867 p^{6} T^{6} - 1153 p^{9} T^{7} + p^{12} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 1472 T + 1805084 T^{2} - 1370871744 T^{3} + 894579577654 T^{4} - 1370871744 p^{3} T^{5} + 1805084 p^{6} T^{6} - 1472 p^{9} T^{7} + p^{12} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 200 T + 364322 T^{2} - 121468265 T^{3} + 278790401066 T^{4} - 121468265 p^{3} T^{5} + 364322 p^{6} T^{6} - 200 p^{9} T^{7} + p^{12} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 1147 T + 1912188 T^{2} + 1347657401 T^{3} + 1180324718270 T^{4} + 1347657401 p^{3} T^{5} + 1912188 p^{6} T^{6} + 1147 p^{9} T^{7} + p^{12} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 908 T + 1048492 T^{2} + 1070426012 T^{3} + 693505308902 T^{4} + 1070426012 p^{3} T^{5} + 1048492 p^{6} T^{6} + 908 p^{9} T^{7} + p^{12} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 1048 T + 1927704 T^{2} - 1122552888 T^{3} + 1371917773086 T^{4} - 1122552888 p^{3} T^{5} + 1927704 p^{6} T^{6} - 1048 p^{9} T^{7} + p^{12} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 1784 T + 2327792 T^{2} + 1523808488 T^{3} + 1318662054974 T^{4} + 1523808488 p^{3} T^{5} + 2327792 p^{6} T^{6} + 1784 p^{9} T^{7} + p^{12} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 2047 T + 2933893 T^{2} - 2857813609 T^{3} + 3031159147844 T^{4} - 2857813609 p^{3} T^{5} + 2933893 p^{6} T^{6} - 2047 p^{9} T^{7} + p^{12} T^{8} \) |
| 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
−6.68981266677302054500238558850, −6.03164704783195438591295746694, −5.99335028184985165141454229017, −5.89356877106471206194902561946, −5.88784016571939726379610531925, −5.29594595302402353850247093017, −5.14321967409647888074247283958, −5.00369738640981915817815614554, −4.95347587085721802077118675885, −4.54008364859338653568598130767, −4.40009114014946322318414811727, −4.04297521290752859998567375464, −3.83593772253296687283727488935, −3.29342218637467650535782256838, −3.27671817268913015900782136382, −3.20838343985474801613661244388, −3.02638265178186521779912879144, −2.73125405530215496146245016214, −2.18441962621242589274718233359, −2.16555958730734190863171700395, −2.06758786712221674805792032915, −1.46773354936662879242766287686, −0.938445260559382823204613178656, −0.878836373911864542103782376266, −0.32015779495147579307099293244,
0.32015779495147579307099293244, 0.878836373911864542103782376266, 0.938445260559382823204613178656, 1.46773354936662879242766287686, 2.06758786712221674805792032915, 2.16555958730734190863171700395, 2.18441962621242589274718233359, 2.73125405530215496146245016214, 3.02638265178186521779912879144, 3.20838343985474801613661244388, 3.27671817268913015900782136382, 3.29342218637467650535782256838, 3.83593772253296687283727488935, 4.04297521290752859998567375464, 4.40009114014946322318414811727, 4.54008364859338653568598130767, 4.95347587085721802077118675885, 5.00369738640981915817815614554, 5.14321967409647888074247283958, 5.29594595302402353850247093017, 5.88784016571939726379610531925, 5.89356877106471206194902561946, 5.99335028184985165141454229017, 6.03164704783195438591295746694, 6.68981266677302054500238558850
