| L(s) = 1 | − 3·3-s − 5-s + 4·7-s − 9-s + 4·11-s + 4·13-s + 3·15-s − 7·17-s − 9·19-s − 12·21-s + 3·23-s − 11·25-s + 12·27-s − 3·29-s − 12·33-s − 4·35-s − 18·37-s − 12·39-s − 14·41-s − 43-s + 45-s − 3·47-s + 10·49-s + 21·51-s + 4·53-s − 4·55-s + 27·57-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 1.51·7-s − 1/3·9-s + 1.20·11-s + 1.10·13-s + 0.774·15-s − 1.69·17-s − 2.06·19-s − 2.61·21-s + 0.625·23-s − 2.19·25-s + 2.30·27-s − 0.557·29-s − 2.08·33-s − 0.676·35-s − 2.95·37-s − 1.92·39-s − 2.18·41-s − 0.152·43-s + 0.149·45-s − 0.437·47-s + 10/7·49-s + 2.94·51-s + 0.549·53-s − 0.539·55-s + 3.57·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 7^{4} \cdot 11^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 2 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | $C_1$ | \( ( 1 - T )^{4} \) |
| 13 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 + p T + 10 T^{2} + 7 p T^{3} + 5 p^{2} T^{4} + 7 p^{2} T^{5} + 10 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + T + 12 T^{2} + 21 T^{3} + 69 T^{4} + 21 p T^{5} + 12 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 7 T + 78 T^{2} + 339 T^{3} + 2037 T^{4} + 339 p T^{5} + 78 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 9 T + 98 T^{2} + 525 T^{3} + 3003 T^{4} + 525 p T^{5} + 98 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 3 T + 36 T^{2} + 45 T^{3} + 463 T^{4} + 45 p T^{5} + 36 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 3 T + 84 T^{2} + 237 T^{3} + 3391 T^{4} + 237 p T^{5} + 84 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 102 T^{2} + 32 T^{3} + 4407 T^{4} + 32 p T^{5} + 102 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 18 T + 220 T^{2} + 1809 T^{3} + 12489 T^{4} + 1809 p T^{5} + 220 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 14 T + 172 T^{2} + 1451 T^{3} + 11121 T^{4} + 1451 p T^{5} + 172 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + T + 109 T^{2} - 68 T^{3} + 5677 T^{4} - 68 p T^{5} + 109 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 3 T + 116 T^{2} + 9 p T^{3} + 7215 T^{4} + 9 p^{2} T^{5} + 116 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 4 T + 3 p T^{2} - 426 T^{3} + 11119 T^{4} - 426 p T^{5} + 3 p^{3} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 7 T + 217 T^{2} + 1240 T^{3} + 18645 T^{4} + 1240 p T^{5} + 217 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 14 T + 200 T^{2} + 1725 T^{3} + 16819 T^{4} + 1725 p T^{5} + 200 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 218 T^{2} + 96 T^{3} + 20235 T^{4} + 96 p T^{5} + 218 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 196 T^{2} - 256 T^{3} + 17830 T^{4} - 256 p T^{5} + 196 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 6 T + 126 T^{2} + 877 T^{3} + 8853 T^{4} + 877 p T^{5} + 126 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 7 T + 167 T^{2} - 1026 T^{3} + 16003 T^{4} - 1026 p T^{5} + 167 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 8 T + 2 p T^{2} - 1127 T^{3} + 17547 T^{4} - 1127 p T^{5} + 2 p^{3} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 9 T + 196 T^{2} - 2057 T^{3} + 21133 T^{4} - 2057 p T^{5} + 196 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 16 T + 438 T^{2} + 4611 T^{3} + 66127 T^{4} + 4611 p T^{5} + 438 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.35595221192020605817445284295, −5.92945468137041436558218819597, −5.89546382167176856196860048711, −5.87746851998544582336927749174, −5.84050222686778956114995703351, −5.23132265863538209459865687915, −5.10930772360059868603346896122, −5.09568830242240205929918727742, −4.91242241679693865770921964797, −4.61919628920518865870972571474, −4.42337069578917425965073042464, −4.20422099648493053318122373227, −4.05763482721341261065026585570, −3.72767002967493550591693392898, −3.50222427698195891622524919792, −3.44568520030755071658694306038, −3.37949161426529445494788376707, −2.78204687161298935847381370150, −2.28062900255824912739795896332, −2.27205913315331035207664366573, −2.20547642319442244622272480343, −1.79420065899808663245050448861, −1.37948058554207456198501490443, −1.26771153294288235159964312295, −1.20182689743775018969054388270, 0, 0, 0, 0,
1.20182689743775018969054388270, 1.26771153294288235159964312295, 1.37948058554207456198501490443, 1.79420065899808663245050448861, 2.20547642319442244622272480343, 2.27205913315331035207664366573, 2.28062900255824912739795896332, 2.78204687161298935847381370150, 3.37949161426529445494788376707, 3.44568520030755071658694306038, 3.50222427698195891622524919792, 3.72767002967493550591693392898, 4.05763482721341261065026585570, 4.20422099648493053318122373227, 4.42337069578917425965073042464, 4.61919628920518865870972571474, 4.91242241679693865770921964797, 5.09568830242240205929918727742, 5.10930772360059868603346896122, 5.23132265863538209459865687915, 5.84050222686778956114995703351, 5.87746851998544582336927749174, 5.89546382167176856196860048711, 5.92945468137041436558218819597, 6.35595221192020605817445284295
