L(s) = 1 | + 16-s + 2·25-s − 4·37-s − 8·41-s + 4·53-s + 4·61-s + 8·101-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
L(s) = 1 | + 16-s + 2·25-s − 4·37-s − 8·41-s + 4·53-s + 4·61-s + 8·101-s + 4·109-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 8·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
Λ(s)=(=((216⋅716⋅178)s/2ΓC(s)8L(s)Λ(1−s)
Λ(s)=(=((216⋅716⋅178)s/2ΓC(s)8L(s)Λ(1−s)
Particular Values
L(21) |
≈ |
1.677833456 |
L(21) |
≈ |
1.677833456 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−T4+T8 |
| 7 | 1 |
| 17 | 1−T4+T8 |
good | 3 | 1−T8+T16 |
| 5 | (1−T2+T4)2(1−T4+T8) |
| 11 | 1−T8+T16 |
| 13 | (1−T)8(1+T)8 |
| 19 | (1−T4+T8)2 |
| 23 | 1−T8+T16 |
| 29 | (1+T2)4(1+T4)2 |
| 31 | 1−T8+T16 |
| 37 | (1+T+T2)4(1−T4+T8) |
| 41 | (1+T)8(1+T4)2 |
| 43 | (1+T4)4 |
| 47 | (1−T2+T4)4 |
| 53 | (1−T+T2)4(1−T2+T4)2 |
| 59 | (1−T4+T8)2 |
| 61 | (1−T+T2)4(1−T4+T8) |
| 67 | (1−T+T2)4(1+T+T2)4 |
| 71 | (1+T8)2 |
| 73 | (1−T2+T4)2(1−T4+T8) |
| 79 | 1−T8+T16 |
| 83 | (1+T4)4 |
| 89 | (1−T2+T4)4 |
| 97 | (1+T2)4(1+T4)2 |
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L(s)=p∏ j=1∏16(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−3.67041184292760459942877887781, −3.62955722604908126287703879846, −3.53494915371445483046303017657, −3.41771687755488308128885370052, −3.34899124692051940482125257461, −3.30949861999462570812819989718, −3.28046304674850454497978475998, −3.03258338465119621035031858123, −2.96917741610359589758718964647, −2.76597752983309755168459772768, −2.46550938726616524475874508777, −2.43011682657594202268918753013, −2.35909583238435316373018166071, −2.11276847089133480192130302580, −2.04036288934781870613153549600, −1.87578948984808099513278947038, −1.86137447689934077097655311697, −1.81659003137354326186323287432, −1.72835739662762071265245408365, −1.17118602317589879789733212688, −1.13907200535183769131176040800, −1.05740136694526897986230506952, −1.05328748996031345716370664575, −0.58539248629331278947778383125, −0.38077843168179938311252986372,
0.38077843168179938311252986372, 0.58539248629331278947778383125, 1.05328748996031345716370664575, 1.05740136694526897986230506952, 1.13907200535183769131176040800, 1.17118602317589879789733212688, 1.72835739662762071265245408365, 1.81659003137354326186323287432, 1.86137447689934077097655311697, 1.87578948984808099513278947038, 2.04036288934781870613153549600, 2.11276847089133480192130302580, 2.35909583238435316373018166071, 2.43011682657594202268918753013, 2.46550938726616524475874508777, 2.76597752983309755168459772768, 2.96917741610359589758718964647, 3.03258338465119621035031858123, 3.28046304674850454497978475998, 3.30949861999462570812819989718, 3.34899124692051940482125257461, 3.41771687755488308128885370052, 3.53494915371445483046303017657, 3.62955722604908126287703879846, 3.67041184292760459942877887781
Plot not available for L-functions of degree greater than 10.