| L(s) = 1 | + 4·3-s − 4·5-s − 4·7-s + 10·9-s − 6·13-s − 16·15-s − 2·17-s + 4·19-s − 16·21-s − 2·23-s + 2·25-s + 20·27-s + 2·29-s − 10·31-s + 16·35-s + 6·37-s − 24·39-s − 8·41-s − 8·43-s − 40·45-s + 4·47-s + 10·49-s − 8·51-s + 2·53-s + 16·57-s − 20·61-s − 40·63-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 1.78·5-s − 1.51·7-s + 10/3·9-s − 1.66·13-s − 4.13·15-s − 0.485·17-s + 0.917·19-s − 3.49·21-s − 0.417·23-s + 2/5·25-s + 3.84·27-s + 0.371·29-s − 1.79·31-s + 2.70·35-s + 0.986·37-s − 3.84·39-s − 1.24·41-s − 1.21·43-s − 5.96·45-s + 0.583·47-s + 10/7·49-s − 1.12·51-s + 0.274·53-s + 2.11·57-s − 2.56·61-s − 5.03·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 7^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_1$ | \( ( 1 + T )^{4} \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $C_2 \wr C_2\wr C_2$ | \( 1 + 4 T + 14 T^{2} + 8 p T^{3} + 82 T^{4} + 8 p^{2} T^{5} + 14 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 + 8 T^{2} + 126 T^{4} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 13 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 40 T^{2} + 138 T^{3} + 606 T^{4} + 138 p T^{5} + 40 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 38 T^{2} + 38 T^{3} + 682 T^{4} + 38 p T^{5} + 38 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr C_2\wr C_2$ | \( 1 + 2 T + 32 T^{2} - 22 T^{3} + 478 T^{4} - 22 p T^{5} + 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 74 T^{2} - 230 T^{3} + 2602 T^{4} - 230 p T^{5} + 74 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr C_2\wr C_2$ | \( 1 + 10 T + 100 T^{2} + 586 T^{3} + 3814 T^{4} + 586 p T^{5} + 100 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr C_2\wr C_2$ | \( 1 - 6 T + 100 T^{2} - 594 T^{3} + 4806 T^{4} - 594 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 43 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 160 T^{2} + 920 T^{3} + 10126 T^{4} + 920 p T^{5} + 160 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 86 T^{2} - 352 T^{3} + 6442 T^{4} - 352 p T^{5} + 86 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr C_2\wr C_2$ | \( 1 - 2 T + 170 T^{2} - 374 T^{3} + 12394 T^{4} - 374 p T^{5} + 170 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 61 | $C_2 \wr C_2\wr C_2$ | \( 1 + 20 T + 292 T^{2} + 2876 T^{3} + 25030 T^{4} + 2876 p T^{5} + 292 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr C_2\wr C_2$ | \( 1 + 12 T + 208 T^{2} + 1308 T^{3} + 15918 T^{4} + 1308 p T^{5} + 208 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 254 T^{2} + 1062 T^{3} + 25698 T^{4} + 1062 p T^{5} + 254 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 124 T^{2} + 260 T^{3} + 5734 T^{4} + 260 p T^{5} + 124 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr C_2\wr C_2$ | \( 1 + 34 T + 688 T^{2} + 9370 T^{3} + 96286 T^{4} + 9370 p T^{5} + 688 p^{2} T^{6} + 34 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr C_2\wr C_2$ | \( 1 + 6 T + 182 T^{2} + 918 T^{3} + 20370 T^{4} + 918 p T^{5} + 182 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr C_2\wr C_2$ | \( 1 + 8 T + 236 T^{2} + 1592 T^{3} + 28678 T^{4} + 1592 p T^{5} + 236 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr C_2\wr C_2$ | \( 1 - 4 T + 184 T^{2} + 308 T^{3} + 15022 T^{4} + 308 p T^{5} + 184 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.21404935799934283145985239894, −5.72368614587354194425737676600, −5.55736296524659912811169574917, −5.52960340248379062106988073036, −5.30501650351753481400603351100, −4.80304998480546675854969600887, −4.80242803545712143328492066120, −4.55047334294945581084038608813, −4.50213785614058648759051459749, −4.08087982906211669939639565398, −4.06632226553919398826360541390, −3.84802982388937379663614476383, −3.68273948717289488096274650509, −3.43770010015367491331071819966, −3.36697073750308463910554158097, −3.13800496448750186935344340141, −2.83209147420988782883424220410, −2.69763847587083420433229180319, −2.47464301986679291392230636650, −2.47317649455846177186770578874, −2.21140940809243773185385226876, −1.57583271015860967607275437963, −1.50520705348365584822377559846, −1.31437385646325511559499352835, −1.12223219384984647949556425657, 0, 0, 0, 0,
1.12223219384984647949556425657, 1.31437385646325511559499352835, 1.50520705348365584822377559846, 1.57583271015860967607275437963, 2.21140940809243773185385226876, 2.47317649455846177186770578874, 2.47464301986679291392230636650, 2.69763847587083420433229180319, 2.83209147420988782883424220410, 3.13800496448750186935344340141, 3.36697073750308463910554158097, 3.43770010015367491331071819966, 3.68273948717289488096274650509, 3.84802982388937379663614476383, 4.06632226553919398826360541390, 4.08087982906211669939639565398, 4.50213785614058648759051459749, 4.55047334294945581084038608813, 4.80242803545712143328492066120, 4.80304998480546675854969600887, 5.30501650351753481400603351100, 5.52960340248379062106988073036, 5.55736296524659912811169574917, 5.72368614587354194425737676600, 6.21404935799934283145985239894
