| L(s) = 1 | + 10·3-s − 6·5-s − 110·7-s + 50·9-s + 300·11-s − 360·13-s − 60·15-s + 960·17-s − 1.10e3·21-s + 810·23-s + 946·25-s − 270·27-s + 836·31-s + 3.00e3·33-s + 660·35-s − 660·37-s − 3.60e3·39-s + 2.96e3·41-s − 3.27e3·43-s − 300·45-s + 2.25e3·47-s + 6.05e3·49-s + 9.60e3·51-s + 1.98e3·53-s − 1.80e3·55-s + 1.58e4·61-s − 5.50e3·63-s + ⋯ |
| L(s) = 1 | + 10/9·3-s − 0.239·5-s − 2.24·7-s + 0.617·9-s + 2.47·11-s − 2.13·13-s − 0.266·15-s + 3.32·17-s − 2.49·21-s + 1.53·23-s + 1.51·25-s − 0.370·27-s + 0.869·31-s + 2.75·33-s + 0.538·35-s − 0.482·37-s − 2.36·39-s + 1.76·41-s − 1.76·43-s − 0.148·45-s + 1.01·47-s + 2.51·49-s + 3.69·51-s + 0.704·53-s − 0.595·55-s + 4.25·61-s − 1.38·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(5.288123832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.288123832\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + 6 T - 182 p T^{2} + 6 p^{4} T^{3} + p^{8} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 10 T + 50 T^{2} + 10 p^{3} T^{3} - 14 p^{6} T^{4} + 10 p^{7} T^{5} + 50 p^{8} T^{6} - 10 p^{12} T^{7} + p^{16} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 + 110 T + 6050 T^{2} + 311190 T^{3} + 15823298 T^{4} + 311190 p^{4} T^{5} + 6050 p^{8} T^{6} + 110 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 150 T + 28882 T^{2} - 150 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 + 360 T + 64800 T^{2} + 14552280 T^{3} + 3127330814 T^{4} + 14552280 p^{4} T^{5} + 64800 p^{8} T^{6} + 360 p^{12} T^{7} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 960 T + 460800 T^{2} - 168098880 T^{3} + 52934858494 T^{4} - 168098880 p^{4} T^{5} + 460800 p^{8} T^{6} - 960 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 79796 T^{2} + 32223922086 T^{4} - 79796 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 810 T + 328050 T^{2} - 264893490 T^{3} + 211669226338 T^{4} - 264893490 p^{4} T^{5} + 328050 p^{8} T^{6} - 810 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 1448516 T^{2} + 1055067026886 T^{4} - 1448516 p^{8} T^{6} + p^{16} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 418 T + 535098 T^{2} - 418 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 + 660 T + 217800 T^{2} + 345245340 T^{3} - 1278103400242 T^{4} + 345245340 p^{4} T^{5} + 217800 p^{8} T^{6} + 660 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 1482 T + 6194578 T^{2} - 1482 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 3270 T + 5346450 T^{2} + 11205966270 T^{3} + 23487235317602 T^{4} + 11205966270 p^{4} T^{5} + 5346450 p^{8} T^{6} + 3270 p^{12} T^{7} + p^{16} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2250 T + 2531250 T^{2} - 7891319250 T^{3} + 22718088596834 T^{4} - 7891319250 p^{4} T^{5} + 2531250 p^{8} T^{6} - 2250 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 1980 T + 1960200 T^{2} - 16406396820 T^{3} + 137161065548878 T^{4} - 16406396820 p^{4} T^{5} + 1960200 p^{8} T^{6} - 1980 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 162508 T^{2} + 44050763806758 T^{4} + 162508 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 - 7914 T + 36788306 T^{2} - 7914 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4810 T + 11568050 T^{2} - 108537019890 T^{3} + 1012520446566818 T^{4} - 108537019890 p^{4} T^{5} + 11568050 p^{8} T^{6} - 4810 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 2994 T + 49877146 T^{2} - 2994 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 14060 T + 98841800 T^{2} + 628627141380 T^{3} + 3731941946353934 T^{4} + 628627141380 p^{4} T^{5} + 98841800 p^{8} T^{6} + 14060 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 127811332 T^{2} + 7111883058262278 T^{4} - 127811332 p^{8} T^{6} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 + 19950 T + 199001250 T^{2} + 1392338210550 T^{3} + 9242910030806018 T^{4} + 1392338210550 p^{4} T^{5} + 199001250 p^{8} T^{6} + 19950 p^{12} T^{7} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 213169796 T^{2} + 19233126298796166 T^{4} - 213169796 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 3180 T + 5056200 T^{2} - 122563753980 T^{3} - 13176145025425522 T^{4} - 122563753980 p^{4} T^{5} + 5056200 p^{8} T^{6} + 3180 p^{12} T^{7} + p^{16} T^{8} \) |
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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
−9.696784251848939608698234178404, −9.650694679541261989283628412060, −9.381054733941201214344835802250, −9.176694338247085513458384233631, −8.711115276497580972522856735762, −8.381170199996124934234582324191, −8.273729438805966196725104214598, −7.45066712306243630914097364699, −7.36224125129933241470438290158, −7.25845620690213648186346584806, −6.70463182443263164554311533739, −6.46798190629044881890046339589, −6.43374111304750948611338136705, −5.39379149772941626320948907157, −5.38602563930738788620073556988, −5.20666224322278793354035684441, −4.07345829085527092129118785733, −4.00390758201125539229914274681, −3.79438483031314217530481161279, −2.93726629250831097591683975557, −2.91178233211997921264726114521, −2.79518618279319741534954751038, −1.63071001472344430951549093619, −0.954782214102309314399035775000, −0.64633490696082808369910821592,
0.64633490696082808369910821592, 0.954782214102309314399035775000, 1.63071001472344430951549093619, 2.79518618279319741534954751038, 2.91178233211997921264726114521, 2.93726629250831097591683975557, 3.79438483031314217530481161279, 4.00390758201125539229914274681, 4.07345829085527092129118785733, 5.20666224322278793354035684441, 5.38602563930738788620073556988, 5.39379149772941626320948907157, 6.43374111304750948611338136705, 6.46798190629044881890046339589, 6.70463182443263164554311533739, 7.25845620690213648186346584806, 7.36224125129933241470438290158, 7.45066712306243630914097364699, 8.273729438805966196725104214598, 8.381170199996124934234582324191, 8.711115276497580972522856735762, 9.176694338247085513458384233631, 9.381054733941201214344835802250, 9.650694679541261989283628412060, 9.696784251848939608698234178404
