| L(s) = 1 | + 3-s − 3·5-s + 3·7-s − 5·9-s − 4·11-s − 5·13-s − 3·15-s − 4·19-s + 3·21-s + 8·23-s − 7·25-s − 9·27-s − 29-s + 7·31-s − 4·33-s − 9·35-s − 16·37-s − 5·39-s − 7·41-s − 5·43-s + 15·45-s + 4·47-s − 16·49-s − 18·53-s + 12·55-s − 4·57-s − 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s + 1.13·7-s − 5/3·9-s − 1.20·11-s − 1.38·13-s − 0.774·15-s − 0.917·19-s + 0.654·21-s + 1.66·23-s − 7/5·25-s − 1.73·27-s − 0.185·29-s + 1.25·31-s − 0.696·33-s − 1.52·35-s − 2.63·37-s − 0.800·39-s − 1.09·41-s − 0.762·43-s + 2.23·45-s + 0.583·47-s − 2.28·49-s − 2.47·53-s + 1.61·55-s − 0.529·57-s − 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 11^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - T + 2 p T^{2} - 2 T^{3} + 19 T^{4} - 2 p T^{5} + 2 p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 3 T + 16 T^{2} + 36 T^{3} + 23 p T^{4} + 36 p T^{5} + 16 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 3 T + 25 T^{2} - 59 T^{3} + 254 T^{4} - 59 p T^{5} + 25 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 5 T + 29 T^{2} + 7 p T^{3} + 322 T^{4} + 7 p^{2} T^{5} + 29 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 30 T^{2} + 36 T^{3} + 642 T^{4} + 36 p T^{5} + 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 85 T^{2} - 382 T^{3} + 2536 T^{4} - 382 p T^{5} + 85 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + T + 101 T^{2} + 51 T^{3} + 4154 T^{4} + 51 p T^{5} + 101 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 7 T + 118 T^{2} - 650 T^{3} + 5395 T^{4} - 650 p T^{5} + 118 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 16 T + 221 T^{2} + 1896 T^{3} + 13680 T^{4} + 1896 p T^{5} + 221 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 7 T + 99 T^{2} + 381 T^{3} + 4244 T^{4} + 381 p T^{5} + 99 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 5 T + 153 T^{2} + 637 T^{3} + 9464 T^{4} + 637 p T^{5} + 153 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 4 T + 76 T^{2} - 356 T^{3} + 4822 T^{4} - 356 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 18 T + 304 T^{2} + 2958 T^{3} + 26542 T^{4} + 2958 p T^{5} + 304 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 213 T^{2} - 48 T^{3} + 18156 T^{4} - 48 p T^{5} + 213 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 6 T + 106 T^{2} + 746 T^{3} + 10106 T^{4} + 746 p T^{5} + 106 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + T + 188 T^{2} - 34 T^{3} + 16225 T^{4} - 34 p T^{5} + 188 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - T + 48 T^{2} - 306 T^{3} - 2731 T^{4} - 306 p T^{5} + 48 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 6 T + 176 T^{2} + 1186 T^{3} + 15006 T^{4} + 1186 p T^{5} + 176 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 4 T + 238 T^{2} - 680 T^{3} + 24922 T^{4} - 680 p T^{5} + 238 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 25 T + 433 T^{2} - 5021 T^{3} + 50710 T^{4} - 5021 p T^{5} + 433 p^{2} T^{6} - 25 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 14 T + 341 T^{2} + 3364 T^{3} + 45484 T^{4} + 3364 p T^{5} + 341 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 36 T + 841 T^{2} + 12772 T^{3} + 148064 T^{4} + 12772 p T^{5} + 841 p^{2} T^{6} + 36 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.67179298870682877758389763966, −6.17365003679573478880966785672, −6.01547549143989926023810877619, −5.92696050198383050435357132902, −5.80318462559899113062054668317, −5.12736021939213026695364150501, −5.12720190096475502861260632067, −5.11582592494549537363088163597, −5.08474661021959087709527718764, −4.74058304777526868570179984938, −4.64187966394914407735500004646, −4.18384171204454978971949980825, −4.03722284797253118274272769439, −3.81616072789217002829507078363, −3.45875577748403795169004308534, −3.40088604129924937321526429237, −3.22988784538388998399881373786, −2.91155060690244138106957142700, −2.59018277994471158634674541494, −2.47096660799521095962961951858, −2.40110342912696371293292860967, −1.93174727282761731305088401382, −1.57789796433160359635634521321, −1.46150558430403383828690775447, −1.15836854510595399727086704546, 0, 0, 0, 0,
1.15836854510595399727086704546, 1.46150558430403383828690775447, 1.57789796433160359635634521321, 1.93174727282761731305088401382, 2.40110342912696371293292860967, 2.47096660799521095962961951858, 2.59018277994471158634674541494, 2.91155060690244138106957142700, 3.22988784538388998399881373786, 3.40088604129924937321526429237, 3.45875577748403795169004308534, 3.81616072789217002829507078363, 4.03722284797253118274272769439, 4.18384171204454978971949980825, 4.64187966394914407735500004646, 4.74058304777526868570179984938, 5.08474661021959087709527718764, 5.11582592494549537363088163597, 5.12720190096475502861260632067, 5.12736021939213026695364150501, 5.80318462559899113062054668317, 5.92696050198383050435357132902, 6.01547549143989926023810877619, 6.17365003679573478880966785672, 6.67179298870682877758389763966
