| L(s) = 1 | − 4·2-s + 10·4-s − 2·5-s − 20·8-s + 8·10-s + 4·11-s + 35·16-s − 4·17-s + 10·19-s − 20·20-s − 16·22-s − 2·23-s + 5·25-s − 4·29-s − 24·31-s − 56·32-s + 16·34-s − 4·37-s − 40·38-s + 40·40-s − 4·43-s + 40·44-s + 8·46-s − 20·50-s − 12·53-s − 8·55-s + 16·58-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 0.894·5-s − 7.07·8-s + 2.52·10-s + 1.20·11-s + 35/4·16-s − 0.970·17-s + 2.29·19-s − 4.47·20-s − 3.41·22-s − 0.417·23-s + 25-s − 0.742·29-s − 4.31·31-s − 9.89·32-s + 2.74·34-s − 0.657·37-s − 6.48·38-s + 6.32·40-s − 0.609·43-s + 6.03·44-s + 1.17·46-s − 2.82·50-s − 1.64·53-s − 1.07·55-s + 2.10·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06372122555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06372122555\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 2 T - T^{2} - 2 p T^{3} - 4 p T^{4} - 2 p^{2} T^{5} - p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2$ | \( ( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^3$ | \( 1 - 2 T^{2} - 165 T^{4} - 2 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 10 T + 43 T^{2} - 10 p T^{3} + 52 p T^{4} - 10 p^{2} T^{5} + 43 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 + 4 T - 22 T^{2} - 80 T^{3} + 139 T^{4} - 80 p T^{5} - 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 37 | $D_4\times C_2$ | \( 1 + 4 T + 34 T^{2} - 368 T^{3} - 1637 T^{4} - 368 p T^{5} + 34 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^3$ | \( 1 + 14 T^{2} - 1485 T^{4} + 14 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 4 T - 50 T^{2} - 80 T^{3} + 1819 T^{4} - 80 p T^{5} - 50 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 12 T + 26 T^{2} + 144 T^{3} + 3483 T^{4} + 144 p T^{5} + 26 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 61 | $D_{4}$ | \( ( 1 + 18 T + 197 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 + 16 T + 174 T^{2} + 16 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 10 T + 143 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 4 T - 110 T^{2} - 80 T^{3} + 9379 T^{4} - 80 p T^{5} - 110 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 6 T + 143 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 2 T - 79 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 24 T + 278 T^{2} - 2880 T^{3} + 29619 T^{4} - 2880 p T^{5} + 278 p^{2} T^{6} - 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 4 T - 158 T^{2} - 80 T^{3} + 19315 T^{4} - 80 p T^{5} - 158 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.38995667115646391692715433815, −6.23098562106624898520382230205, −6.02472323959520825532774661001, −5.81578362272830370391322064065, −5.70226449315397610458849061647, −5.34955447816144983921965212469, −5.10884364664703236546607619173, −4.81116534970986262043132583878, −4.56437875071428856707851565317, −4.43983029884359598304602986064, −4.23899127375800743890859745677, −3.58135631603707100344680741783, −3.46458638948626067155197858001, −3.44374718276081407306528964037, −3.38535058999238196475350437225, −2.87415436333564596489701880808, −2.84492031309446567306733428800, −2.34477638617022320976235271018, −1.92140828772393633274759738553, −1.68580527057421435156772691276, −1.57339249104882401931354646301, −1.43288977544691183617403259661, −1.12182342406409323123365894295, −0.38830461814407498132323967983, −0.12429021424867357496684424390,
0.12429021424867357496684424390, 0.38830461814407498132323967983, 1.12182342406409323123365894295, 1.43288977544691183617403259661, 1.57339249104882401931354646301, 1.68580527057421435156772691276, 1.92140828772393633274759738553, 2.34477638617022320976235271018, 2.84492031309446567306733428800, 2.87415436333564596489701880808, 3.38535058999238196475350437225, 3.44374718276081407306528964037, 3.46458638948626067155197858001, 3.58135631603707100344680741783, 4.23899127375800743890859745677, 4.43983029884359598304602986064, 4.56437875071428856707851565317, 4.81116534970986262043132583878, 5.10884364664703236546607619173, 5.34955447816144983921965212469, 5.70226449315397610458849061647, 5.81578362272830370391322064065, 6.02472323959520825532774661001, 6.23098562106624898520382230205, 6.38995667115646391692715433815
