| L(s) = 1 | + 26·4-s − 110·7-s − 332·13-s + 420·16-s − 884·19-s − 286·25-s − 2.86e3·28-s + 802·31-s + 1.00e3·37-s − 4.86e3·43-s + 4.27e3·49-s − 8.63e3·52-s − 1.44e4·61-s + 4.26e3·64-s + 7.09e3·67-s + 1.24e3·73-s − 2.29e4·76-s + 6.31e3·79-s + 3.65e4·91-s − 2.52e4·97-s − 7.43e3·100-s − 1.79e4·103-s − 2.56e4·109-s − 4.62e4·112-s + 2.06e4·121-s + 2.08e4·124-s + 127-s + ⋯ |
| L(s) = 1 | + 13/8·4-s − 2.24·7-s − 1.96·13-s + 1.64·16-s − 2.44·19-s − 0.457·25-s − 3.64·28-s + 0.834·31-s + 0.734·37-s − 2.62·43-s + 1.77·49-s − 3.19·52-s − 3.88·61-s + 1.04·64-s + 1.57·67-s + 0.233·73-s − 3.97·76-s + 1.01·79-s + 4.41·91-s − 2.68·97-s − 0.743·100-s − 1.69·103-s − 2.15·109-s − 3.68·112-s + 1.40·121-s + 1.35·124-s + 6.20e−5·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59049 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.01903947036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01903947036\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 13 p T^{2} + p^{8} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 286 T^{2} + p^{8} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 55 T + p^{4} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 20618 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 166 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 97058 T^{2} + p^{8} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 442 T + p^{4} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 125782 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 818038 T^{2} + p^{8} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 401 T + p^{4} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 503 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 5160746 T^{2} + p^{8} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 2431 T + p^{4} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 8721026 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 12130562 T^{2} + p^{8} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 7098122 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 7225 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 3545 T + p^{4} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 40273706 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 623 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3158 T + p^{4} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 91114946 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 6241898 T^{2} + p^{8} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 12646 T + p^{4} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.20632275266140911465366261631, −11.00916505112718252255229836658, −10.85641128766470304876646822213, −10.13085245258772100824666090962, −9.827996557869809652026829949683, −9.611189869583600499663975532162, −8.822578025861744778482251589397, −8.071821260752944670561625799042, −7.68677814840022312772189168652, −6.83490436182087181688902643485, −6.70683947752961425980750185961, −6.37638712571007749449551151677, −5.87352541015846113754847384813, −4.95924501058312302950692523107, −4.24664270905624579529838344553, −3.38540084934081476333065962992, −2.80723937478662728941079753466, −2.43246142690907532731138705826, −1.68460358159235603079688521457, −0.04029655677978634387954486585,
0.04029655677978634387954486585, 1.68460358159235603079688521457, 2.43246142690907532731138705826, 2.80723937478662728941079753466, 3.38540084934081476333065962992, 4.24664270905624579529838344553, 4.95924501058312302950692523107, 5.87352541015846113754847384813, 6.37638712571007749449551151677, 6.70683947752961425980750185961, 6.83490436182087181688902643485, 7.68677814840022312772189168652, 8.071821260752944670561625799042, 8.822578025861744778482251589397, 9.611189869583600499663975532162, 9.827996557869809652026829949683, 10.13085245258772100824666090962, 10.85641128766470304876646822213, 11.00916505112718252255229836658, 12.20632275266140911465366261631
