| L(s) = 1 | − 4·3-s − 8·7-s + 8·9-s + 24·19-s + 32·21-s + 80·25-s + 4·27-s + 56·31-s + 32·37-s − 136·43-s − 128·49-s − 96·57-s − 160·61-s − 64·63-s + 280·67-s − 80·73-s − 320·75-s + 408·79-s − 30·81-s − 224·93-s + 96·97-s − 488·103-s + 160·109-s − 128·111-s + 480·121-s + 127-s + 544·129-s + ⋯ |
| L(s) = 1 | − 4/3·3-s − 8/7·7-s + 8/9·9-s + 1.26·19-s + 1.52·21-s + 16/5·25-s + 4/27·27-s + 1.80·31-s + 0.864·37-s − 3.16·43-s − 2.61·49-s − 1.68·57-s − 2.62·61-s − 1.01·63-s + 4.17·67-s − 1.09·73-s − 4.26·75-s + 5.16·79-s − 0.370·81-s − 2.40·93-s + 0.989·97-s − 4.73·103-s + 1.46·109-s − 1.15·111-s + 3.96·121-s + 0.00787·127-s + 4.21·129-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{56} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9112187135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9112187135\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 4 T + 8 T^{2} - 4 T^{3} - 22 p T^{4} - 4 p^{2} T^{5} + 8 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| good | 5 | \( 1 - 16 p T^{2} + 3004 T^{4} - 18672 p T^{6} + 2634054 T^{8} - 18672 p^{5} T^{10} + 3004 p^{8} T^{12} - 16 p^{13} T^{14} + p^{16} T^{16} \) |
| 7 | \( ( 1 + 4 T + 88 T^{2} + 220 T^{3} + 3646 T^{4} + 220 p^{2} T^{5} + 88 p^{4} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 11 | \( 1 - 480 T^{2} + 125212 T^{4} - 22531104 T^{6} + 3078264198 T^{8} - 22531104 p^{4} T^{10} + 125212 p^{8} T^{12} - 480 p^{12} T^{14} + p^{16} T^{16} \) |
| 13 | \( ( 1 + 316 T^{2} + 256 T^{3} + 57510 T^{4} + 256 p^{2} T^{5} + 316 p^{4} T^{6} + p^{8} T^{8} )^{2} \) |
| 17 | \( 1 - 1064 T^{2} + 654556 T^{4} - 282654744 T^{6} + 92440824774 T^{8} - 282654744 p^{4} T^{10} + 654556 p^{8} T^{12} - 1064 p^{12} T^{14} + p^{16} T^{16} \) |
| 19 | \( ( 1 - 12 T + 664 T^{2} + 4588 T^{3} + 120414 T^{4} + 4588 p^{2} T^{5} + 664 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 23 | \( 1 - 1896 T^{2} + 2093020 T^{4} - 1543350744 T^{6} + 914302437702 T^{8} - 1543350744 p^{4} T^{10} + 2093020 p^{8} T^{12} - 1896 p^{12} T^{14} + p^{16} T^{16} \) |
| 29 | \( 1 - 3024 T^{2} + 4188220 T^{4} - 3460574000 T^{6} + 2571955018182 T^{8} - 3460574000 p^{4} T^{10} + 4188220 p^{8} T^{12} - 3024 p^{12} T^{14} + p^{16} T^{16} \) |
| 31 | \( ( 1 - 28 T + 3016 T^{2} - 63204 T^{3} + 3999966 T^{4} - 63204 p^{2} T^{5} + 3016 p^{4} T^{6} - 28 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 37 | \( ( 1 - 16 T + 1948 T^{2} + 25232 T^{3} + 1096870 T^{4} + 25232 p^{2} T^{5} + 1948 p^{4} T^{6} - 16 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 41 | \( 1 - 2472 T^{2} + 3015004 T^{4} - 2001918872 T^{6} + 1122734027334 T^{8} - 2001918872 p^{4} T^{10} + 3015004 p^{8} T^{12} - 2472 p^{12} T^{14} + p^{16} T^{16} \) |
| 43 | \( ( 1 + 68 T + 6808 T^{2} + 284700 T^{3} + 17420958 T^{4} + 284700 p^{2} T^{5} + 6808 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 47 | \( 1 - 12296 T^{2} + 72283420 T^{4} - 268133584440 T^{6} + 697058148522822 T^{8} - 268133584440 p^{4} T^{10} + 72283420 p^{8} T^{12} - 12296 p^{12} T^{14} + p^{16} T^{16} \) |
| 53 | \( 1 + 304 T^{2} + 25898428 T^{4} + 6393839056 T^{6} + 288293861465158 T^{8} + 6393839056 p^{4} T^{10} + 25898428 p^{8} T^{12} + 304 p^{12} T^{14} + p^{16} T^{16} \) |
| 59 | \( 1 - 19008 T^{2} + 181109212 T^{4} - 1092163345344 T^{6} + 4537129454477574 T^{8} - 1092163345344 p^{4} T^{10} + 181109212 p^{8} T^{12} - 19008 p^{12} T^{14} + p^{16} T^{16} \) |
| 61 | \( ( 1 + 80 T + 12700 T^{2} + 630832 T^{3} + 62237030 T^{4} + 630832 p^{2} T^{5} + 12700 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 67 | \( ( 1 - 140 T + 18856 T^{2} - 1550068 T^{3} + 122202110 T^{4} - 1550068 p^{2} T^{5} + 18856 p^{4} T^{6} - 140 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 71 | \( 1 - 17640 T^{2} + 102948316 T^{4} - 54971476568 T^{6} - 1448379553007034 T^{8} - 54971476568 p^{4} T^{10} + 102948316 p^{8} T^{12} - 17640 p^{12} T^{14} + p^{16} T^{16} \) |
| 73 | \( ( 1 + 40 T + 15772 T^{2} + 446872 T^{3} + 110402182 T^{4} + 446872 p^{2} T^{5} + 15772 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 79 | \( ( 1 - 204 T + 32488 T^{2} - 3143092 T^{3} + 288852126 T^{4} - 3143092 p^{2} T^{5} + 32488 p^{4} T^{6} - 204 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
| 83 | \( 1 - 41696 T^{2} + 833769628 T^{4} - 10294236337952 T^{6} + 85511640874023430 T^{8} - 10294236337952 p^{4} T^{10} + 833769628 p^{8} T^{12} - 41696 p^{12} T^{14} + p^{16} T^{16} \) |
| 89 | \( 1 - 17736 T^{2} + 318384412 T^{4} - 3335315895288 T^{6} + 31854516109881798 T^{8} - 3335315895288 p^{4} T^{10} + 318384412 p^{8} T^{12} - 17736 p^{12} T^{14} + p^{16} T^{16} \) |
| 97 | \( ( 1 - 48 T + 29500 T^{2} - 1080784 T^{3} + 386488710 T^{4} - 1080784 p^{2} T^{5} + 29500 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.92691258768584356577976278133, −4.69079525820332987185123725016, −4.58878539366391634238581197189, −4.39979561489589857347963658636, −4.33493821574499734565531826597, −4.15883191078827555893645497124, −3.62401126547305621980653735101, −3.55135948872783259554338886855, −3.50518323985027702640348836970, −3.50181635712018353166146597611, −3.43421964385639732640469960015, −3.15086397439557809777194827379, −2.78771713069567420328667420746, −2.75136326761587394029992012940, −2.54866881538509597707144581350, −2.49307680086248986191771769704, −2.32342130797280380126038348209, −1.72378244215110354799361106732, −1.71135737550822014512358310904, −1.43724910744318389921781160570, −1.20821677972574415288188597650, −0.967947560833860947786823882329, −0.74184620650847777060335736870, −0.53299343288937070833272280916, −0.13909824807290844820878247311,
0.13909824807290844820878247311, 0.53299343288937070833272280916, 0.74184620650847777060335736870, 0.967947560833860947786823882329, 1.20821677972574415288188597650, 1.43724910744318389921781160570, 1.71135737550822014512358310904, 1.72378244215110354799361106732, 2.32342130797280380126038348209, 2.49307680086248986191771769704, 2.54866881538509597707144581350, 2.75136326761587394029992012940, 2.78771713069567420328667420746, 3.15086397439557809777194827379, 3.43421964385639732640469960015, 3.50181635712018353166146597611, 3.50518323985027702640348836970, 3.55135948872783259554338886855, 3.62401126547305621980653735101, 4.15883191078827555893645497124, 4.33493821574499734565531826597, 4.39979561489589857347963658636, 4.58878539366391634238581197189, 4.69079525820332987185123725016, 4.92691258768584356577976278133
Plot not available for L-functions of degree greater than 10.
