| L(s) = 1 | + 2·2-s + 2·3-s + 3·4-s + 4·6-s − 2·7-s + 2·8-s + 5·9-s − 4·11-s + 6·12-s + 8·13-s − 4·14-s + 4·17-s + 10·18-s − 4·21-s − 8·22-s − 2·23-s + 4·24-s + 16·26-s + 10·27-s − 6·28-s − 4·29-s + 12·31-s − 6·32-s − 8·33-s + 8·34-s + 15·36-s + 16·39-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 1.63·6-s − 0.755·7-s + 0.707·8-s + 5/3·9-s − 1.20·11-s + 1.73·12-s + 2.21·13-s − 1.06·14-s + 0.970·17-s + 2.35·18-s − 0.872·21-s − 1.70·22-s − 0.417·23-s + 0.816·24-s + 3.13·26-s + 1.92·27-s − 1.13·28-s − 0.742·29-s + 2.15·31-s − 1.06·32-s − 1.39·33-s + 1.37·34-s + 5/2·36-s + 2.56·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.940227099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.940227099\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - p T + T^{2} + p T^{3} - 3 T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{4} T^{7} + p^{4} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 T - T^{2} + 2 T^{3} + 4 T^{4} + 2 p T^{5} - p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 4 T - 2 T^{2} - 16 T^{3} + 27 T^{4} - 16 p T^{5} - 2 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 4 T - 14 T^{2} + 16 T^{3} + 339 T^{4} + 16 p T^{5} - 14 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^3$ | \( 1 - 30 T^{2} + 539 T^{4} - 30 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 2 T - 41 T^{2} - 2 T^{3} + 1404 T^{4} - 2 p T^{5} - 41 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{4} \) |
| 31 | $C_2^2$ | \( ( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 - p T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 8 T - 50 T^{2} + 64 T^{3} + 6795 T^{4} + 64 p T^{5} - 50 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 8 T + 2 T^{2} + 448 T^{3} - 3413 T^{4} + 448 p T^{5} + 2 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 6 T - 23 T^{2} + 378 T^{3} - 2436 T^{4} + 378 p T^{5} - 23 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 22 T + 231 T^{2} - 2618 T^{3} + 27092 T^{4} - 2618 p T^{5} + 231 p^{2} T^{6} - 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 4 T - 126 T^{2} + 16 T^{3} + 13667 T^{4} + 16 p T^{5} - 126 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 24 T + 282 T^{2} + 3264 T^{3} + 34691 T^{4} + 3264 p T^{5} + 282 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 - 6 T - 119 T^{2} + 138 T^{3} + 12900 T^{4} + 138 p T^{5} - 119 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−9.461877721834180559844678091312, −8.724372711696904480656613429132, −8.685819166400673554071254659739, −8.609283637749227736513072157875, −8.227603736651881293425160547535, −8.101552133121464899516032950856, −7.61376866886425519954578900839, −7.40473957804389206003968003085, −6.93150157492033961969785268208, −6.65994446408115717263248017646, −6.65628991099025667557913204667, −6.27453371851719097167305360405, −5.75327466049295665616040899848, −5.72203833901601097691517284851, −5.18494296427381168925440005700, −5.00133818761332780826369219697, −4.59585027681082808139750843716, −4.02062679772013825390819254827, −3.86847158550969055579115645997, −3.40547581335048268628475731018, −3.29905384986511619460563763128, −2.90723323392972382989691767426, −2.54355492331874252910419348469, −1.75796855752372390685593674987, −1.43696065403407671941128734596,
1.43696065403407671941128734596, 1.75796855752372390685593674987, 2.54355492331874252910419348469, 2.90723323392972382989691767426, 3.29905384986511619460563763128, 3.40547581335048268628475731018, 3.86847158550969055579115645997, 4.02062679772013825390819254827, 4.59585027681082808139750843716, 5.00133818761332780826369219697, 5.18494296427381168925440005700, 5.72203833901601097691517284851, 5.75327466049295665616040899848, 6.27453371851719097167305360405, 6.65628991099025667557913204667, 6.65994446408115717263248017646, 6.93150157492033961969785268208, 7.40473957804389206003968003085, 7.61376866886425519954578900839, 8.101552133121464899516032950856, 8.227603736651881293425160547535, 8.609283637749227736513072157875, 8.685819166400673554071254659739, 8.724372711696904480656613429132, 9.461877721834180559844678091312
