L(s) = 1 | + 4·7-s + 11-s + 2·13-s + 3·17-s + 4·19-s − 6·23-s + 6·29-s − 5·31-s + 2·37-s − 3·41-s − 8·43-s + 6·47-s + 9·49-s − 12·53-s − 6·59-s − 4·61-s − 5·67-s + 12·71-s + 2·73-s + 4·77-s + 10·79-s + 9·83-s + 18·89-s + 8·91-s − 97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | + 1.51·7-s + 0.301·11-s + 0.554·13-s + 0.727·17-s + 0.917·19-s − 1.25·23-s + 1.11·29-s − 0.898·31-s + 0.328·37-s − 0.468·41-s − 1.21·43-s + 0.875·47-s + 9/7·49-s − 1.64·53-s − 0.781·59-s − 0.512·61-s − 0.610·67-s + 1.42·71-s + 0.234·73-s + 0.455·77-s + 1.12·79-s + 0.987·83-s + 1.90·89-s + 0.838·91-s − 0.101·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
Λ(s)=(=(118800s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(118800s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
3.778335631 |
L(21) |
≈ |
3.778335631 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1 |
| 11 | 1−T |
good | 7 | 1−4T+pT2 |
| 13 | 1−2T+pT2 |
| 17 | 1−3T+pT2 |
| 19 | 1−4T+pT2 |
| 23 | 1+6T+pT2 |
| 29 | 1−6T+pT2 |
| 31 | 1+5T+pT2 |
| 37 | 1−2T+pT2 |
| 41 | 1+3T+pT2 |
| 43 | 1+8T+pT2 |
| 47 | 1−6T+pT2 |
| 53 | 1+12T+pT2 |
| 59 | 1+6T+pT2 |
| 61 | 1+4T+pT2 |
| 67 | 1+5T+pT2 |
| 71 | 1−12T+pT2 |
| 73 | 1−2T+pT2 |
| 79 | 1−10T+pT2 |
| 83 | 1−9T+pT2 |
| 89 | 1−18T+pT2 |
| 97 | 1+T+pT2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−13.74540224369684, −13.21641322451448, −12.38696326711440, −12.06335452519689, −11.72851527803853, −11.13850093430593, −10.76194389172191, −10.23225020667047, −9.661591095635382, −9.165470782382390, −8.553163364134243, −8.086407162849880, −7.729019757041997, −7.321352964515868, −6.403852433019532, −6.136165997410227, −5.373663060150516, −4.931139653013213, −4.539698401419140, −3.662598855632647, −3.420171203146060, −2.474051113354234, −1.762515033328033, −1.369962923142672, −0.6182185328294570,
0.6182185328294570, 1.369962923142672, 1.762515033328033, 2.474051113354234, 3.420171203146060, 3.662598855632647, 4.539698401419140, 4.931139653013213, 5.373663060150516, 6.136165997410227, 6.403852433019532, 7.321352964515868, 7.729019757041997, 8.086407162849880, 8.553163364134243, 9.165470782382390, 9.661591095635382, 10.23225020667047, 10.76194389172191, 11.13850093430593, 11.72851527803853, 12.06335452519689, 12.38696326711440, 13.21641322451448, 13.74540224369684