L(s) = 1 | + 18·3-s − 98·7-s + 243·9-s + 100·11-s + 540·13-s + 1.15e3·17-s + 2.94e3·19-s − 1.76e3·21-s + 2.68e3·23-s − 1.73e3·25-s + 2.91e3·27-s + 996·29-s + 2.61e3·31-s + 1.80e3·33-s − 3.49e3·37-s + 9.72e3·39-s + 1.70e4·41-s + 1.36e4·43-s + 1.18e4·47-s + 7.20e3·49-s + 2.07e4·51-s + 3.75e4·53-s + 5.29e4·57-s − 6.32e3·59-s + 3.97e4·61-s − 2.38e4·63-s − 2.73e4·67-s + ⋯ |
L(s) = 1 | + 1.15·3-s − 0.755·7-s + 9-s + 0.249·11-s + 0.886·13-s + 0.966·17-s + 1.87·19-s − 0.872·21-s + 1.05·23-s − 0.554·25-s + 0.769·27-s + 0.219·29-s + 0.488·31-s + 0.287·33-s − 0.419·37-s + 1.02·39-s + 1.58·41-s + 1.12·43-s + 0.781·47-s + 3/7·49-s + 1.11·51-s + 1.83·53-s + 2.16·57-s − 0.236·59-s + 1.36·61-s − 0.755·63-s − 0.745·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28224 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(5.805229612\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.805229612\) |
\(L(\frac{7}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
good | 5 | $C_2^2$ | \( 1 + 1734 T^{2} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 100 T + 320086 T^{2} - 100 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 540 T + 526462 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 1152 T + 3130846 T^{2} - 1152 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2944 T + 5655798 T^{2} - 2944 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2684 T + 3830734 T^{2} - 2684 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 996 T + 37205902 T^{2} - 996 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 2616 T + 41609662 T^{2} - 2616 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 3492 T + 72298414 T^{2} + 3492 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 17016 T + 272987742 T^{2} - 17016 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 13640 T + 279761990 T^{2} - 13640 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 11832 T + 202334814 T^{2} - 11832 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 37548 T + 17523254 p T^{2} - 37548 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 6320 T + 780841414 T^{2} + 6320 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 39764 T + 1977385070 T^{2} - 39764 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 27376 T - 46144618 T^{2} + 27376 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 35460 T + 3145043502 T^{2} - 35460 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 64188 T + 4387114438 T^{2} - 64188 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 110088 T + 8439555358 T^{2} + 110088 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 5896 T + 7776870614 T^{2} + 5896 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 167160 T + 16792608382 T^{2} - 167160 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 11876 T + 16694014454 T^{2} + 11876 p^{5} T^{3} + p^{10} T^{4} \) |
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
−12.00966253842578642818859549387, −11.95265075213618678779511576109, −11.01380061730178881994259595637, −10.62852697835868727213198955101, −9.805412413462515376952272122002, −9.711733347828222598616793761568, −8.970940462729748614648497991231, −8.822912633980712027772947173624, −7.86958745091277688507382198088, −7.64454775023428536718660796607, −7.00984979063330933664896188521, −6.43318494405745576205044510504, −5.61756169943708083214830773851, −5.21718185677927462142237574311, −3.90876434420072511251789907978, −3.85986523544438859760547088380, −2.91509801000489828593235243029, −2.56819974292697023958284381970, −1.23388709406404342786805464838, −0.871748704563070905586264274815,
0.871748704563070905586264274815, 1.23388709406404342786805464838, 2.56819974292697023958284381970, 2.91509801000489828593235243029, 3.85986523544438859760547088380, 3.90876434420072511251789907978, 5.21718185677927462142237574311, 5.61756169943708083214830773851, 6.43318494405745576205044510504, 7.00984979063330933664896188521, 7.64454775023428536718660796607, 7.86958745091277688507382198088, 8.822912633980712027772947173624, 8.970940462729748614648497991231, 9.711733347828222598616793761568, 9.805412413462515376952272122002, 10.62852697835868727213198955101, 11.01380061730178881994259595637, 11.95265075213618678779511576109, 12.00966253842578642818859549387