| L(s) = 1 | + 2-s − 4·3-s − 3·4-s − 4·6-s − 4·7-s − 3·8-s + 10·9-s + 2·11-s + 12·12-s − 4·13-s − 4·14-s + 4·16-s + 5·17-s + 10·18-s − 15·19-s + 16·21-s + 2·22-s + 8·23-s + 12·24-s − 4·26-s − 20·27-s + 12·28-s − 5·29-s − 15·31-s + 3·32-s − 8·33-s + 5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s − 3/2·4-s − 1.63·6-s − 1.51·7-s − 1.06·8-s + 10/3·9-s + 0.603·11-s + 3.46·12-s − 1.10·13-s − 1.06·14-s + 16-s + 1.21·17-s + 2.35·18-s − 3.44·19-s + 3.49·21-s + 0.426·22-s + 1.66·23-s + 2.44·24-s − 0.784·26-s − 3.84·27-s + 2.26·28-s − 0.928·29-s − 2.69·31-s + 0.530·32-s − 1.39·33-s + 0.857·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 - T + p^{2} T^{2} - p^{2} T^{3} + 9 T^{4} - p^{3} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2 T + 3 p T^{2} - 6 p T^{3} + 487 T^{4} - 6 p^{2} T^{5} + 3 p^{3} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 5 T + 58 T^{2} - 179 T^{3} + 1305 T^{4} - 179 p T^{5} + 58 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 15 T + 141 T^{2} + 904 T^{3} + 4497 T^{4} + 904 p T^{5} + 141 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 8 T + 94 T^{2} - 464 T^{3} + 3135 T^{4} - 464 p T^{5} + 94 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 5 T + 48 T^{2} + 291 T^{3} + 2263 T^{4} + 291 p T^{5} + 48 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 15 T + 200 T^{2} + 1533 T^{3} + 10517 T^{4} + 1533 p T^{5} + 200 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 12 T + 128 T^{2} - 915 T^{3} + 5979 T^{4} - 915 p T^{5} + 128 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 5 T + 100 T^{2} + 669 T^{3} + 5033 T^{4} + 669 p T^{5} + 100 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 11 T + 169 T^{2} - 1226 T^{3} + 10939 T^{4} - 1226 p T^{5} + 169 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 4 T + 77 T^{2} + 712 T^{3} + 3001 T^{4} + 712 p T^{5} + 77 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 15 T + 246 T^{2} - 2337 T^{3} + 20413 T^{4} - 2337 p T^{5} + 246 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 11 T + 174 T^{2} + 1533 T^{3} + 15199 T^{4} + 1533 p T^{5} + 174 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 20 T + 348 T^{2} + 3767 T^{3} + 35009 T^{4} + 3767 p T^{5} + 348 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 2 T + 75 T^{2} + 576 T^{3} + 2797 T^{4} + 576 p T^{5} + 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 20 T + 306 T^{2} - 3288 T^{3} + 32139 T^{4} - 3288 p T^{5} + 306 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6 T + 68 T^{2} - 39 T^{3} + 5069 T^{4} - 39 p T^{5} + 68 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 7 T + 213 T^{2} + 766 T^{3} + 19505 T^{4} + 766 p T^{5} + 213 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2 T + 82 T^{2} - 213 T^{3} + 15107 T^{4} - 213 p T^{5} + 82 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 232 T^{2} + 2127 T^{3} + 26639 T^{4} + 2127 p T^{5} + 232 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5 T + 168 T^{2} + 2027 T^{3} + 16505 T^{4} + 2027 p T^{5} + 168 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.01872323124004378330805692938, −5.61881764783868880715102406426, −5.55246441354956261289027183381, −5.48893290700624066826637770765, −5.32362501107215506230246750870, −4.89904349174151183376399603146, −4.83838812752241727436730769494, −4.78722593812153336300267770752, −4.52897140773896486943736291697, −4.24839247565297147291591378081, −4.11517932066554777454231792778, −4.08756548205072641643894716196, −3.92775553765426675247511933265, −3.45060760235694540203889213293, −3.42844565735932152817942188726, −3.40412297277270160229191302025, −2.92197478532194785176105133575, −2.58290062257086065721385950684, −2.39773845041420189910664970023, −2.13560263932550365352612355973, −2.01530423009745876687990225332, −1.60079247423531015889019944674, −1.12417606255947401130589873568, −0.995338874430046744089833466660, −0.927306254623728863092151512968, 0, 0, 0, 0,
0.927306254623728863092151512968, 0.995338874430046744089833466660, 1.12417606255947401130589873568, 1.60079247423531015889019944674, 2.01530423009745876687990225332, 2.13560263932550365352612355973, 2.39773845041420189910664970023, 2.58290062257086065721385950684, 2.92197478532194785176105133575, 3.40412297277270160229191302025, 3.42844565735932152817942188726, 3.45060760235694540203889213293, 3.92775553765426675247511933265, 4.08756548205072641643894716196, 4.11517932066554777454231792778, 4.24839247565297147291591378081, 4.52897140773896486943736291697, 4.78722593812153336300267770752, 4.83838812752241727436730769494, 4.89904349174151183376399603146, 5.32362501107215506230246750870, 5.48893290700624066826637770765, 5.55246441354956261289027183381, 5.61881764783868880715102406426, 6.01872323124004378330805692938
