L(s) = 1 | + 10·5-s − 4·7-s + 20·11-s − 70·13-s − 90·17-s − 140·19-s + 192·23-s − 25·25-s − 134·29-s + 100·31-s − 40·35-s + 170·37-s + 110·41-s − 532·43-s + 56·47-s − 327·49-s − 430·53-s + 200·55-s − 20·59-s − 270·61-s − 700·65-s + 524·67-s + 80·71-s + 330·73-s − 80·77-s + 1.06e3·79-s − 1.18e3·83-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 0.215·7-s + 0.548·11-s − 1.49·13-s − 1.28·17-s − 1.69·19-s + 1.74·23-s − 1/5·25-s − 0.858·29-s + 0.579·31-s − 0.193·35-s + 0.755·37-s + 0.419·41-s − 1.88·43-s + 0.173·47-s − 0.953·49-s − 1.11·53-s + 0.490·55-s − 0.0441·59-s − 0.566·61-s − 1.33·65-s + 0.955·67-s + 0.133·71-s + 0.529·73-s − 0.118·77-s + 1.50·79-s − 1.57·83-s + ⋯ |
Λ(s)=(=(576s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(576s/2ΓC(s+3/2)L(s)−Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
good | 5 | 1−2pT+p3T2 |
| 7 | 1+4T+p3T2 |
| 11 | 1−20T+p3T2 |
| 13 | 1+70T+p3T2 |
| 17 | 1+90T+p3T2 |
| 19 | 1+140T+p3T2 |
| 23 | 1−192T+p3T2 |
| 29 | 1+134T+p3T2 |
| 31 | 1−100T+p3T2 |
| 37 | 1−170T+p3T2 |
| 41 | 1−110T+p3T2 |
| 43 | 1+532T+p3T2 |
| 47 | 1−56T+p3T2 |
| 53 | 1+430T+p3T2 |
| 59 | 1+20T+p3T2 |
| 61 | 1+270T+p3T2 |
| 67 | 1−524T+p3T2 |
| 71 | 1−80T+p3T2 |
| 73 | 1−330T+p3T2 |
| 79 | 1−1060T+p3T2 |
| 83 | 1+1188T+p3T2 |
| 89 | 1+1274T+p3T2 |
| 97 | 1+590T+p3T2 |
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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
−9.694397508797212442745035212872, −9.245377962400217435059977358078, −8.208332681649884755425609288332, −6.89199337660583853317715284780, −6.41716780902006901716200848148, −5.16741838372930162984051879029, −4.30632769635815019718880272138, −2.74634064417202034913997095541, −1.81110578588999287003461155390, 0,
1.81110578588999287003461155390, 2.74634064417202034913997095541, 4.30632769635815019718880272138, 5.16741838372930162984051879029, 6.41716780902006901716200848148, 6.89199337660583853317715284780, 8.208332681649884755425609288332, 9.245377962400217435059977358078, 9.694397508797212442745035212872