| L(s) = 1 | + 4·2-s + 6·3-s + 11·4-s + 24·6-s + 20·7-s + 18·8-s + 4·9-s + 4·11-s + 66·12-s + 80·14-s + 24·16-s + 30·17-s + 16·18-s − 10·19-s + 120·21-s + 16·22-s − 30·23-s + 108·24-s − 48·27-s + 220·28-s − 2·29-s − 20·31-s + 40·32-s + 24·33-s + 120·34-s + 44·36-s + 38·37-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·3-s + 11/4·4-s + 4·6-s + 20/7·7-s + 9/4·8-s + 4/9·9-s + 4/11·11-s + 11/2·12-s + 40/7·14-s + 3/2·16-s + 1.76·17-s + 8/9·18-s − 0.526·19-s + 40/7·21-s + 8/11·22-s − 1.30·23-s + 9/2·24-s − 1.77·27-s + 55/7·28-s − 0.0689·29-s − 0.645·31-s + 5/4·32-s + 8/11·33-s + 3.52·34-s + 11/9·36-s + 1.02·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(46.13791363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(46.13791363\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p^{2} T + 5 T^{2} + 3 p T^{3} - 31 T^{4} + 3 p^{3} T^{5} + 5 p^{4} T^{6} - p^{8} T^{7} + p^{8} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 p T + 32 T^{2} - 40 p T^{3} + 427 T^{4} - 40 p^{3} T^{5} + 32 p^{4} T^{6} - 2 p^{7} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 20 T + 164 T^{2} - 636 T^{3} + 1871 T^{4} - 636 p^{2} T^{5} + 164 p^{4} T^{6} - 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 200 T^{2} + 960 T^{3} + 17471 T^{4} + 960 p^{2} T^{5} + 200 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 30 T + 109 T^{2} - 6390 T^{3} + 282060 T^{4} - 6390 p^{2} T^{5} + 109 p^{4} T^{6} - 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 10 T + 74 T^{2} + 1752 T^{3} - 96625 T^{4} + 1752 p^{2} T^{5} + 74 p^{4} T^{6} + 10 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 30 T - 356 T^{2} + 5940 T^{3} + 821595 T^{4} + 5940 p^{2} T^{5} - 356 p^{4} T^{6} + 30 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 2 T - 1571 T^{2} - 214 T^{3} + 1769980 T^{4} - 214 p^{2} T^{5} - 1571 p^{4} T^{6} + 2 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 + 20 T + 200 T^{2} + 18300 T^{3} + 1672334 T^{4} + 18300 p^{2} T^{5} + 200 p^{4} T^{6} + 20 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 38 T + 41 p T^{2} + 52158 T^{3} - 1571608 T^{4} + 52158 p^{2} T^{5} + 41 p^{5} T^{6} - 38 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 100 T + 3461 T^{2} + 7272 T^{3} - 4096804 T^{4} + 7272 p^{2} T^{5} + 3461 p^{4} T^{6} - 100 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $C_2^3$ | \( 1 - 998 T^{2} - 2422797 T^{4} - 998 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 68 T + 2312 T^{2} - 148716 T^{3} + 9565454 T^{4} - 148716 p^{2} T^{5} + 2312 p^{4} T^{6} - 68 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 4814 T^{2} + 12616659 T^{4} - 4814 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 164 T + 6980 T^{2} + 551844 T^{3} - 68910913 T^{4} + 551844 p^{2} T^{5} + 6980 p^{4} T^{6} - 164 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 124 T + 5413 T^{2} + 312604 T^{3} + 28201432 T^{4} + 312604 p^{2} T^{5} + 5413 p^{4} T^{6} + 124 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 170 T + 10706 T^{2} + 134496 T^{3} - 16049425 T^{4} + 134496 p^{2} T^{5} + 10706 p^{4} T^{6} + 170 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 86 T + 4658 T^{2} + 258336 T^{3} + 1457087 T^{4} + 258336 p^{2} T^{5} + 4658 p^{4} T^{6} + 86 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 58 T + 1682 T^{2} - 201144 T^{3} + 20590727 T^{4} - 201144 p^{2} T^{5} + 1682 p^{4} T^{6} - 58 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 20 T + 7290 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 188 T + 17672 T^{2} - 1935084 T^{3} + 200304482 T^{4} - 1935084 p^{2} T^{5} + 17672 p^{4} T^{6} - 188 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 110 T + 12050 T^{2} - 1194180 T^{3} + 87746159 T^{4} - 1194180 p^{2} T^{5} + 12050 p^{4} T^{6} - 110 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 146 T + 13250 T^{2} + 1056876 T^{3} + 58176719 T^{4} + 1056876 p^{2} T^{5} + 13250 p^{4} T^{6} + 146 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.118457700415294272671309750117, −8.011081904864059280461810189008, −7.57447314446836938643214649205, −7.52818519600205263785211336126, −7.51125404454494083365657841164, −7.03820543988924174430114208697, −6.30971668414257459969868298673, −6.28806494152267641826896973027, −6.14612607435310282997340183108, −5.60462254556254356240575248797, −5.52120761455596250884022833453, −5.22285932357240795417696970405, −5.17718321961048515021767233068, −4.47647874769398939143661797268, −4.24175593550368344883698065000, −4.10212119971811778641107401862, −3.91738110732460318556756332605, −3.43787167151867080021803595552, −3.01968102188935734951451491473, −2.72948404376771182773260207413, −2.64216616365783932381048504615, −2.15811524893472269621915804861, −1.93020398229162609867671501007, −1.36681850672607975807255644242, −0.889280544878358500846078123277,
0.889280544878358500846078123277, 1.36681850672607975807255644242, 1.93020398229162609867671501007, 2.15811524893472269621915804861, 2.64216616365783932381048504615, 2.72948404376771182773260207413, 3.01968102188935734951451491473, 3.43787167151867080021803595552, 3.91738110732460318556756332605, 4.10212119971811778641107401862, 4.24175593550368344883698065000, 4.47647874769398939143661797268, 5.17718321961048515021767233068, 5.22285932357240795417696970405, 5.52120761455596250884022833453, 5.60462254556254356240575248797, 6.14612607435310282997340183108, 6.28806494152267641826896973027, 6.30971668414257459969868298673, 7.03820543988924174430114208697, 7.51125404454494083365657841164, 7.52818519600205263785211336126, 7.57447314446836938643214649205, 8.011081904864059280461810189008, 8.118457700415294272671309750117
