| L(s) = 1 | + 7-s + 3·11-s − 2·13-s − 9·17-s − 4·19-s + 3·23-s − 9·29-s + 2·31-s + 37-s + 9·41-s − 8·43-s − 12·47-s − 9·49-s − 12·53-s − 15·59-s − 61-s − 11·67-s + 12·71-s + 10·73-s + 3·77-s − 7·79-s − 12·83-s − 3·89-s − 2·91-s − 5·97-s + 12·101-s + 7·103-s + ⋯ |
| L(s) = 1 | + 0.377·7-s + 0.904·11-s − 0.554·13-s − 2.18·17-s − 0.917·19-s + 0.625·23-s − 1.67·29-s + 0.359·31-s + 0.164·37-s + 1.40·41-s − 1.21·43-s − 1.75·47-s − 9/7·49-s − 1.64·53-s − 1.95·59-s − 0.128·61-s − 1.34·67-s + 1.42·71-s + 1.17·73-s + 0.341·77-s − 0.787·79-s − 1.31·83-s − 0.317·89-s − 0.209·91-s − 0.507·97-s + 1.19·101-s + 0.689·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - T + 10 T^{2} - 31 T^{3} + 43 T^{4} - 31 p T^{5} + 10 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 3 T + 29 T^{2} - 54 T^{3} + 369 T^{4} - 54 p T^{5} + 29 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 2 T + 34 T^{2} + 77 T^{3} + 589 T^{4} + 77 p T^{5} + 34 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 9 T + 80 T^{2} + 441 T^{3} + 2115 T^{4} + 441 p T^{5} + 80 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 4 T + 49 T^{2} + 148 T^{3} + 1117 T^{4} + 148 p T^{5} + 49 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 3 T + p T^{2} + 36 T^{3} + 27 T^{4} + 36 p T^{5} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 9 T + 56 T^{2} + 117 T^{3} + 405 T^{4} + 117 p T^{5} + 56 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 2 T + 106 T^{2} - 149 T^{3} + 4675 T^{4} - 149 p T^{5} + 106 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - T + 109 T^{2} - 88 T^{3} + 5425 T^{4} - 88 p T^{5} + 109 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 9 T + 119 T^{2} - 702 T^{3} + 6477 T^{4} - 702 p T^{5} + 119 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 8 T + 118 T^{2} + 608 T^{3} + 6487 T^{4} + 608 p T^{5} + 118 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 137 T^{2} + 1116 T^{3} + 7461 T^{4} + 1116 p T^{5} + 137 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 12 T + 134 T^{2} + 1287 T^{3} + 11367 T^{4} + 1287 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 15 T + 122 T^{2} + 27 T^{3} - 2637 T^{4} + 27 p T^{5} + 122 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + T + 169 T^{2} - 128 T^{3} + 12859 T^{4} - 128 p T^{5} + 169 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 11 T + 184 T^{2} + 1613 T^{3} + 18535 T^{4} + 1613 p T^{5} + 184 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 12 T + 212 T^{2} - 1665 T^{3} + 19293 T^{4} - 1665 p T^{5} + 212 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 10 T + 280 T^{2} - 1897 T^{3} + 29707 T^{4} - 1897 p T^{5} + 280 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 7 T + p T^{2} + 10 p T^{3} + 15493 T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 12 T + 236 T^{2} + 2295 T^{3} + 29097 T^{4} + 2295 p T^{5} + 236 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 3 T + 113 T^{2} + 1314 T^{3} + 10185 T^{4} + 1314 p T^{5} + 113 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5 T + 253 T^{2} + 788 T^{3} + 32275 T^{4} + 788 p T^{5} + 253 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.96385005057689939075625308476, −5.64834149698759137638269209933, −5.36457339488635670262404931385, −5.22066622811373495571706164804, −5.19370950570824984585935498538, −4.78318570469792800809332639777, −4.70975836051447053359465680049, −4.68393222673971360752365387681, −4.41853573805097903892239266886, −4.22833331457940241719346385267, −3.92403998176683022503862785656, −3.86631087657258397063155654324, −3.75864857148106477619265408302, −3.28563530431361841409850384394, −3.24832872956972015290753434734, −3.00168405481693189380235743242, −2.85140874850316102006042805612, −2.34447782051954629856768543659, −2.24089355602346076763201143035, −2.21857892492807091460646387458, −2.10496249293632736723417664644, −1.40575248999649833310577122070, −1.35187458612823354250604345573, −1.23146979048232082764371199717, −1.22383645751313171064137343074, 0, 0, 0, 0,
1.22383645751313171064137343074, 1.23146979048232082764371199717, 1.35187458612823354250604345573, 1.40575248999649833310577122070, 2.10496249293632736723417664644, 2.21857892492807091460646387458, 2.24089355602346076763201143035, 2.34447782051954629856768543659, 2.85140874850316102006042805612, 3.00168405481693189380235743242, 3.24832872956972015290753434734, 3.28563530431361841409850384394, 3.75864857148106477619265408302, 3.86631087657258397063155654324, 3.92403998176683022503862785656, 4.22833331457940241719346385267, 4.41853573805097903892239266886, 4.68393222673971360752365387681, 4.70975836051447053359465680049, 4.78318570469792800809332639777, 5.19370950570824984585935498538, 5.22066622811373495571706164804, 5.36457339488635670262404931385, 5.64834149698759137638269209933, 5.96385005057689939075625308476
