| L(s) = 1 | − 18·3-s − 8·4-s − 66·5-s − 70·7-s + 189·9-s − 162·11-s + 144·12-s + 1.18e3·15-s − 204·17-s − 444·19-s + 528·20-s + 1.26e3·21-s + 312·23-s + 1.31e3·25-s − 1.45e3·27-s + 560·28-s + 2.72e3·29-s − 3.78e3·31-s + 2.91e3·33-s + 4.62e3·35-s − 1.51e3·36-s + 1.39e3·37-s − 632·43-s + 1.29e3·44-s − 1.24e4·45-s − 7.89e3·47-s + 2.40e3·49-s + ⋯ |
| L(s) = 1 | − 2·3-s − 1/2·4-s − 2.63·5-s − 1.42·7-s + 7/3·9-s − 1.33·11-s + 12-s + 5.27·15-s − 0.705·17-s − 1.22·19-s + 1.31·20-s + 20/7·21-s + 0.589·23-s + 2.10·25-s − 2·27-s + 5/7·28-s + 3.23·29-s − 3.93·31-s + 2.67·33-s + 3.77·35-s − 7/6·36-s + 1.01·37-s − 0.341·43-s + 0.669·44-s − 6.15·45-s − 3.57·47-s + 49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3111696 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1142454476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1142454476\) |
| \(L(3)\) |
|
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 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T + p^{3} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 10 p T + 51 p^{2} T^{2} + 10 p^{5} T^{3} + p^{8} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 66 T + 3041 T^{2} + 104874 T^{3} + 3041796 T^{4} + 104874 p^{4} T^{5} + 3041 p^{8} T^{6} + 66 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 162 T - 9311 T^{2} + 1016226 T^{3} + 665560740 T^{4} + 1016226 p^{4} T^{5} - 9311 p^{8} T^{6} + 162 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 50980 T^{2} + 1849726854 T^{4} - 50980 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 12 p T + 542 p^{2} T^{2} + 5928 p^{3} T^{3} + 174387 p^{4} T^{4} + 5928 p^{7} T^{5} + 542 p^{10} T^{6} + 12 p^{13} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 444 T + 315038 T^{2} + 110700744 T^{3} + 53743544787 T^{4} + 110700744 p^{4} T^{5} + 315038 p^{8} T^{6} + 444 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 312 T + 262414 T^{2} + 226122624 T^{3} - 78303722685 T^{4} + 226122624 p^{4} T^{5} + 262414 p^{8} T^{6} - 312 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1362 T + 1874795 T^{2} - 1362 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 3786 T + 7637801 T^{2} + 10827464034 T^{3} + 11738480198292 T^{4} + 10827464034 p^{4} T^{5} + 7637801 p^{8} T^{6} + 3786 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 1396 T - 2268278 T^{2} - 654405712 T^{3} + 10619007407539 T^{4} - 654405712 p^{4} T^{5} - 2268278 p^{8} T^{6} - 1396 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 2317228 T^{2} + 5437495988070 T^{4} - 2317228 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 316 T - 3534234 T^{2} + 316 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 168 p T + 32962802 T^{2} + 2046329040 p T^{3} + 225964882234371 T^{4} + 2046329040 p^{5} T^{5} + 32962802 p^{8} T^{6} + 168 p^{13} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 1038 T - 12178631 T^{2} - 2620832706 T^{3} + 104962282583700 T^{4} - 2620832706 p^{4} T^{5} - 12178631 p^{8} T^{6} + 1038 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 966 T + 14295473 T^{2} + 13508950686 T^{3} + 52502722474692 T^{4} + 13508950686 p^{4} T^{5} + 14295473 p^{8} T^{6} + 966 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 5088 T + 37832738 T^{2} - 148587357120 T^{3} + 780615710940387 T^{4} - 148587357120 p^{4} T^{5} + 37832738 p^{8} T^{6} - 5088 p^{12} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 14600 T + 121457326 T^{2} - 750446307200 T^{3} + 3707899788833635 T^{4} - 750446307200 p^{4} T^{5} + 121457326 p^{8} T^{6} - 14600 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 4848 T + 52412546 T^{2} + 4848 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 22584 T + 263311922 T^{2} - 2107077488880 T^{3} + 12726401415363651 T^{4} - 2107077488880 p^{4} T^{5} + 263311922 p^{8} T^{6} - 22584 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 3974 T + 5512993 T^{2} + 268723783546 T^{3} - 2026562936732828 T^{4} + 268723783546 p^{4} T^{5} + 5512993 p^{8} T^{6} - 3974 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1452506 p T^{2} + 7047387074675283 T^{4} - 1452506 p^{9} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 33156 T + 547930046 T^{2} + 6017480251704 T^{3} + 51993281156793267 T^{4} + 6017480251704 p^{4} T^{5} + 547930046 p^{8} T^{6} + 33156 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 347883310 T^{2} + 45930093674217747 T^{4} - 347883310 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−11.21113520241450686464224636099, −11.07583073519812671282175951655, −10.99754582046008135712481247391, −10.22762298009413030304431728961, −10.05579358654567040896560908666, −9.701404869765280392685935704482, −9.443278270233122208696910349472, −8.616198391398499759997982866571, −8.513461754460515562636713109758, −8.095716347713537995701107822048, −7.81954727127925057209467155950, −7.36786516011411637452826745338, −6.84024858086050808473345619818, −6.62621496979631768334268126376, −6.51380775269341677071313114921, −5.72389449553646269190787369851, −5.36751440368906582812486783957, −4.88653222506638658314314640790, −4.51235265272485890990695384286, −4.15999634413400173504192926364, −3.43155639102114000304224415141, −3.39742501952175080326639774109, −2.18237728783842648973651337013, −0.44197560749307898514498397904, −0.35861722902111823352446220949,
0.35861722902111823352446220949, 0.44197560749307898514498397904, 2.18237728783842648973651337013, 3.39742501952175080326639774109, 3.43155639102114000304224415141, 4.15999634413400173504192926364, 4.51235265272485890990695384286, 4.88653222506638658314314640790, 5.36751440368906582812486783957, 5.72389449553646269190787369851, 6.51380775269341677071313114921, 6.62621496979631768334268126376, 6.84024858086050808473345619818, 7.36786516011411637452826745338, 7.81954727127925057209467155950, 8.095716347713537995701107822048, 8.513461754460515562636713109758, 8.616198391398499759997982866571, 9.443278270233122208696910349472, 9.701404869765280392685935704482, 10.05579358654567040896560908666, 10.22762298009413030304431728961, 10.99754582046008135712481247391, 11.07583073519812671282175951655, 11.21113520241450686464224636099
