| L(s) = 1 | − 2-s − 3·3-s − 4-s + 4·5-s + 3·6-s − 4·7-s − 2·8-s − 9-s − 4·10-s − 2·11-s + 3·12-s − 7·13-s + 4·14-s − 12·15-s + 3·16-s − 17-s + 18-s − 4·20-s + 12·21-s + 2·22-s + 2·23-s + 6·24-s + 10·25-s + 7·26-s + 11·27-s + 4·28-s + 29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s − 1/2·4-s + 1.78·5-s + 1.22·6-s − 1.51·7-s − 0.707·8-s − 1/3·9-s − 1.26·10-s − 0.603·11-s + 0.866·12-s − 1.94·13-s + 1.06·14-s − 3.09·15-s + 3/4·16-s − 0.242·17-s + 0.235·18-s − 0.894·20-s + 2.61·21-s + 0.426·22-s + 0.417·23-s + 1.22·24-s + 2·25-s + 1.37·26-s + 2.11·27-s + 0.755·28-s + 0.185·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{4} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 19 | | \( 1 \) |
| good | 2 | $C_2 \wr S_4$ | \( 1 + T + p T^{2} + 5 T^{3} + 3 p T^{4} + 5 p T^{5} + p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 3 | $C_2 \wr S_4$ | \( 1 + p T + 10 T^{2} + 22 T^{3} + 44 T^{4} + 22 p T^{5} + 10 p^{2} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 + 4 T + 27 T^{2} + 69 T^{3} + 272 T^{4} + 69 p T^{5} + 27 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 19 T^{2} + 47 T^{3} + 179 T^{4} + 47 p T^{5} + 19 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 7 T + 59 T^{2} + 250 T^{3} + 1180 T^{4} + 250 p T^{5} + 59 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + T + 38 T^{2} + 4 p T^{3} + 822 T^{4} + 4 p^{2} T^{5} + 38 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 2 T + 75 T^{2} - 145 T^{3} + 2398 T^{4} - 145 p T^{5} + 75 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - T + 53 T^{2} - 281 T^{3} + 1251 T^{4} - 281 p T^{5} + 53 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 57 T^{2} + 5 T^{3} + 2675 T^{4} + 5 p T^{5} + 57 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 2 T + 117 T^{2} - 99 T^{3} + 5802 T^{4} - 99 p T^{5} + 117 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 8 T + 77 T^{2} - 13 T^{3} + 714 T^{4} - 13 p T^{5} + 77 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - T + 74 T^{2} + 68 T^{3} + 3460 T^{4} + 68 p T^{5} + 74 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 133 T^{2} + 763 T^{3} + 5768 T^{4} + 763 p T^{5} + 133 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 5 T + 126 T^{2} - 568 T^{3} + 7684 T^{4} - 568 p T^{5} + 126 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 5 T + 111 T^{2} - 385 T^{3} + 8011 T^{4} - 385 p T^{5} + 111 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 114 T^{2} + 88 T^{3} + 9515 T^{4} + 88 p T^{5} + 114 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 4 T + 232 T^{2} + 828 T^{3} + 22174 T^{4} + 828 p T^{5} + 232 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 20 T + 375 T^{2} + 4165 T^{3} + 42925 T^{4} + 4165 p T^{5} + 375 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 20 T + 387 T^{2} + 57 p T^{3} + 44118 T^{4} + 57 p^{2} T^{5} + 387 p^{2} T^{6} + 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 17 T + 388 T^{2} + 4009 T^{3} + 48638 T^{4} + 4009 p T^{5} + 388 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - T + 270 T^{2} - 194 T^{3} + 31408 T^{4} - 194 p T^{5} + 270 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 11 T + 266 T^{2} + 1549 T^{3} + 27690 T^{4} + 1549 p T^{5} + 266 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + T + 122 T^{2} + 1096 T^{3} + 12292 T^{4} + 1096 p T^{5} + 122 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−6.75438075247896739999333605444, −6.71949400603587628508691324235, −6.65450871120510131109952998104, −6.22903257339959368728716007293, −6.07655783799814774466721230009, −6.02240943842882371402363523957, −5.72764270394373176427265263489, −5.66047252117505559364157363350, −5.35808491183678254085268779575, −5.29481583955594560029017412512, −5.02479799624286547433525875574, −4.74859551964247702417842764991, −4.69172659591841668354774507392, −4.34752989850164000978516432661, −3.82395968936054226756220164693, −3.82110276363225392706712260288, −3.29489228731112373624026158128, −3.10298327831670570272976978284, −2.69431846628208284459932820864, −2.55784431112748628696764641358, −2.54905147759130759838841325908, −2.49056635855927055048534884776, −1.55021993786131977251769176065, −1.40657514029036047204288276562, −1.10034973340201255740875042074, 0, 0, 0, 0,
1.10034973340201255740875042074, 1.40657514029036047204288276562, 1.55021993786131977251769176065, 2.49056635855927055048534884776, 2.54905147759130759838841325908, 2.55784431112748628696764641358, 2.69431846628208284459932820864, 3.10298327831670570272976978284, 3.29489228731112373624026158128, 3.82110276363225392706712260288, 3.82395968936054226756220164693, 4.34752989850164000978516432661, 4.69172659591841668354774507392, 4.74859551964247702417842764991, 5.02479799624286547433525875574, 5.29481583955594560029017412512, 5.35808491183678254085268779575, 5.66047252117505559364157363350, 5.72764270394373176427265263489, 6.02240943842882371402363523957, 6.07655783799814774466721230009, 6.22903257339959368728716007293, 6.65450871120510131109952998104, 6.71949400603587628508691324235, 6.75438075247896739999333605444
