| L(s) = 1 | + 8·2-s + 16·4-s − 54·5-s + 74·7-s − 128·8-s + 378·9-s − 432·10-s − 78·11-s + 1.10e3·13-s + 592·14-s − 1.02e3·16-s + 984·17-s + 3.02e3·18-s − 3.28e3·19-s − 864·20-s − 624·22-s − 5.53e3·23-s + 6.11e3·25-s + 8.84e3·26-s + 1.18e3·28-s − 3.89e3·29-s + 4.71e3·31-s − 2.04e3·32-s + 7.87e3·34-s − 3.99e3·35-s + 6.04e3·36-s − 9.59e3·37-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.965·5-s + 0.570·7-s − 0.707·8-s + 14/9·9-s − 1.36·10-s − 0.194·11-s + 1.81·13-s + 0.807·14-s − 16-s + 0.825·17-s + 2.19·18-s − 2.08·19-s − 0.482·20-s − 0.274·22-s − 2.18·23-s + 1.95·25-s + 2.56·26-s + 0.285·28-s − 0.859·29-s + 0.881·31-s − 0.353·32-s + 1.16·34-s − 0.551·35-s + 7/9·36-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104976 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.121449862\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.121449862\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{2} T + p^{4} T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 - 14 p^{3} T^{2} + p^{10} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 54 T - 3199 T^{2} - 1458 p T^{3} + 740604 p^{2} T^{4} - 1458 p^{6} T^{5} - 3199 p^{10} T^{6} + 54 p^{15} T^{7} + p^{20} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 74 T - 29291 T^{2} - 85322 T^{3} + 834233908 T^{4} - 85322 p^{5} T^{5} - 29291 p^{10} T^{6} - 74 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 78 T + 81845 T^{2} - 31033314 T^{3} - 21177529764 T^{4} - 31033314 p^{5} T^{5} + 81845 p^{10} T^{6} + 78 p^{15} T^{7} + p^{20} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 1106 T + 261241 T^{2} - 242666354 T^{3} + 333396924028 T^{4} - 242666354 p^{5} T^{5} + 261241 p^{10} T^{6} - 1106 p^{15} T^{7} + p^{20} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 492 T + 991654 T^{2} - 492 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 + 1640 T + 2303382 T^{2} + 1640 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 + 5538 T + 10264397 T^{2} + 41714215218 T^{3} + 177540370655268 T^{4} + 41714215218 p^{5} T^{5} + 10264397 p^{10} T^{6} + 5538 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 3894 T - 26328655 T^{2} + 1828595142 T^{3} + 989854392537516 T^{4} + 1828595142 p^{5} T^{5} - 26328655 p^{10} T^{6} + 3894 p^{15} T^{7} + p^{20} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 4718 T + 308941 T^{2} + 166581818242 T^{3} - 975071466725036 T^{4} + 166581818242 p^{5} T^{5} + 308941 p^{10} T^{6} - 4718 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 4796 T - 2933298 T^{2} + 4796 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 15354 T - 54861079 T^{2} - 904258368522 T^{3} + 43716860789439108 T^{4} - 904258368522 p^{5} T^{5} - 54861079 p^{10} T^{6} - 15354 p^{15} T^{7} + p^{20} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 32858 T + 531117661 T^{2} - 8362808427386 T^{3} + 120556834830700108 T^{4} - 8362808427386 p^{5} T^{5} + 531117661 p^{10} T^{6} - 32858 p^{15} T^{7} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 24954 T + 21728789 T^{2} - 3550537792602 T^{3} + 165756186376684068 T^{4} - 3550537792602 p^{5} T^{5} + 21728789 p^{10} T^{6} - 24954 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 16332 T + 896230798 T^{2} + 16332 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 21966 T - 1047077995 T^{2} - 2190769191198 T^{3} + 1483553202105261756 T^{4} - 2190769191198 p^{5} T^{5} - 1047077995 p^{10} T^{6} - 21966 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 50 p T - 1573942703 T^{2} + 5297369950 p T^{3} + 1785465604069576108 T^{4} + 5297369950 p^{6} T^{5} - 1573942703 p^{10} T^{6} - 50 p^{16} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 36758 T - 1083763235 T^{2} + 145570685710 p T^{3} + 508296971311036 p^{2} T^{4} + 145570685710 p^{6} T^{5} - 1083763235 p^{10} T^{6} - 36758 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 73848 T + 4273554814 T^{2} + 73848 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 51188 T + 63530214 p T^{2} + 51188 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 14926 T - 5879056307 T^{2} - 780197169890 T^{3} + 27078673587607710964 T^{4} - 780197169890 p^{5} T^{5} - 5879056307 p^{10} T^{6} + 14926 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 90762 T - 1695108259 T^{2} - 186494818454154 T^{3} + 50696650939413455868 T^{4} - 186494818454154 p^{5} T^{5} - 1695108259 p^{10} T^{6} - 90762 p^{15} T^{7} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 9300 T + 3189231862 T^{2} + 9300 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 30262 T - 7152864215 T^{2} - 275566608895610 T^{3} - 16503442419968593676 T^{4} - 275566608895610 p^{5} T^{5} - 7152864215 p^{10} T^{6} + 30262 p^{15} T^{7} + p^{20} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.07345333670908913539755920854, −12.69221572029100091838100060267, −12.62411716956373630162670812521, −12.23125130720910284910479610614, −11.83865182996358570874042275611, −11.45150232959813451227415718304, −10.86555498261630164020485736686, −10.57559409097599858330115099374, −10.42152559117605619823078976369, −9.893578548057804709635961400549, −8.931740363194924856166255454556, −8.915779079601797956667019923252, −8.486321866727081155140317189704, −7.82096121188874813614081769066, −7.48744271895944263411487098725, −7.08629286298606661071889121808, −6.13022856964328002006875292662, −6.08349614267157565472955749215, −5.50062149911938647193123872670, −4.34834546863170708381835743615, −4.28784310012076385842983112758, −4.13378035640076703048240740136, −3.27599549842234091842542519150, −2.08851336967836819239291254926, −0.874351368839317877182743688301,
0.874351368839317877182743688301, 2.08851336967836819239291254926, 3.27599549842234091842542519150, 4.13378035640076703048240740136, 4.28784310012076385842983112758, 4.34834546863170708381835743615, 5.50062149911938647193123872670, 6.08349614267157565472955749215, 6.13022856964328002006875292662, 7.08629286298606661071889121808, 7.48744271895944263411487098725, 7.82096121188874813614081769066, 8.486321866727081155140317189704, 8.915779079601797956667019923252, 8.931740363194924856166255454556, 9.893578548057804709635961400549, 10.42152559117605619823078976369, 10.57559409097599858330115099374, 10.86555498261630164020485736686, 11.45150232959813451227415718304, 11.83865182996358570874042275611, 12.23125130720910284910479610614, 12.62411716956373630162670812521, 12.69221572029100091838100060267, 13.07345333670908913539755920854
