L(s) = 1 | − 2·3-s + 4-s − 7-s + 9-s − 2·12-s + 13-s + 16-s + 2·21-s − 8·25-s − 28-s − 31-s + 36-s + 3·37-s − 2·39-s − 5·43-s − 2·48-s + 49-s + 52-s + 2·61-s − 63-s + 67-s − 73-s + 16·75-s − 2·79-s + 2·84-s − 91-s + 2·93-s + ⋯ |
L(s) = 1 | − 2·3-s + 4-s − 7-s + 9-s − 2·12-s + 13-s + 16-s + 2·21-s − 8·25-s − 28-s − 31-s + 36-s + 3·37-s − 2·39-s − 5·43-s − 2·48-s + 49-s + 52-s + 2·61-s − 63-s + 67-s − 73-s + 16·75-s − 2·79-s + 2·84-s − 91-s + 2·93-s + ⋯ |
Λ(s)=(=((38⋅78⋅318)s/2ΓC(s)8L(s)Λ(1−s)
Λ(s)=(=((38⋅78⋅318)s/2ΓC(s)8L(s)Λ(1−s)
Particular Values
L(21) |
≈ |
0.07830178488 |
L(21) |
≈ |
0.07830178488 |
L(1) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | (1+T+T2+T3+T4)2 |
| 7 | 1+T−T3−T4−T5+T7+T8 |
| 31 | 1+T−T3−T4−T5+T7+T8 |
good | 2 | (1−T+T3−T4+T5−T7+T8)(1+T−T3−T4−T5+T7+T8) |
| 5 | (1+T2)8 |
| 11 | 1+T2−T6−T8−T10+T14+T16 |
| 13 | (1−T+T2−T3+T4)2(1+T−T3−T4−T5+T7+T8) |
| 17 | (1−T+T2−T3+T4)2(1+T+T2+T3+T4)2 |
| 19 | (1−T+T3−T4+T5−T7+T8)(1+T−T3−T4−T5+T7+T8) |
| 23 | 1+T2−T6−T8−T10+T14+T16 |
| 29 | (1−T2+T4−T6+T8)2 |
| 37 | (1−T+T2−T3+T4)2(1−T+T3−T4+T5−T7+T8) |
| 41 | 1+T2−T6−T8−T10+T14+T16 |
| 43 | (1+T+T2)4(1+T−T3−T4−T5+T7+T8) |
| 47 | 1+T2−T6−T8−T10+T14+T16 |
| 53 | 1+T2−T6−T8−T10+T14+T16 |
| 59 | (1−T2+T4−T6+T8)2 |
| 61 | (1−T+T3−T4+T5−T7+T8)2 |
| 67 | (1−T+T2−T3+T4)2(1+T−T3−T4−T5+T7+T8) |
| 71 | (1−T+T3−T4+T5−T7+T8)(1+T−T3−T4−T5+T7+T8) |
| 73 | (1+T+T2+T3+T4)2(1−T+T3−T4+T5−T7+T8) |
| 79 | (1+T+T2)4(1−T+T2−T3+T4)2 |
| 83 | (1−T+T3−T4+T5−T7+T8)(1+T−T3−T4−T5+T7+T8) |
| 89 | (1−T+T3−T4+T5−T7+T8)(1+T−T3−T4−T5+T7+T8) |
| 97 | (1+T+T2)4(1+T−T3−T4−T5+T7+T8) |
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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
−5.12575272077948439768520545609, −4.55436590653883558039446470200, −4.46722183974658321599090360415, −4.44855560745142449581421036003, −4.33796815582767955250492189968, −4.06429979470476895484924199903, −4.02534695338517359404518679576, −3.97336186919728612823989764482, −3.74885049385110226276987567578, −3.70168269131654433246116271288, −3.49673447100127983341624560276, −3.43964702152219192528045791733, −2.99451864133082306711841498771, −2.97357804455718438916504704335, −2.90816659943790793453485417432, −2.84712714102701571400045086997, −2.36188041589498580223174952639, −2.30491786140114385542705935895, −1.94372654216960986925790702935, −1.87042952854743141367267594926, −1.78775132148771652639996393074, −1.74363270228082841656787205857, −1.37858950131948021447132864807, −1.10266904532117267014344145942, −0.35256402371821093890312635013,
0.35256402371821093890312635013, 1.10266904532117267014344145942, 1.37858950131948021447132864807, 1.74363270228082841656787205857, 1.78775132148771652639996393074, 1.87042952854743141367267594926, 1.94372654216960986925790702935, 2.30491786140114385542705935895, 2.36188041589498580223174952639, 2.84712714102701571400045086997, 2.90816659943790793453485417432, 2.97357804455718438916504704335, 2.99451864133082306711841498771, 3.43964702152219192528045791733, 3.49673447100127983341624560276, 3.70168269131654433246116271288, 3.74885049385110226276987567578, 3.97336186919728612823989764482, 4.02534695338517359404518679576, 4.06429979470476895484924199903, 4.33796815582767955250492189968, 4.44855560745142449581421036003, 4.46722183974658321599090360415, 4.55436590653883558039446470200, 5.12575272077948439768520545609
Plot not available for L-functions of degree greater than 10.