L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 2·13-s − 15-s + 2·17-s − 4·19-s + 25-s + 27-s + 2·29-s + 4·33-s + 10·37-s + 2·39-s + 10·41-s − 4·43-s − 45-s + 8·47-s − 7·49-s + 2·51-s + 10·53-s − 4·55-s − 4·57-s + 4·59-s + 2·61-s − 2·65-s − 12·67-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.258·15-s + 0.485·17-s − 0.917·19-s + 1/5·25-s + 0.192·27-s + 0.371·29-s + 0.696·33-s + 1.64·37-s + 0.320·39-s + 1.56·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s + 0.280·51-s + 1.37·53-s − 0.539·55-s − 0.529·57-s + 0.520·59-s + 0.256·61-s − 0.248·65-s − 1.46·67-s + ⋯ |
Λ(s)=(=(960s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(960s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.980751817 |
L(21) |
≈ |
1.980751817 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−T |
| 5 | 1+T |
good | 7 | 1+pT2 |
| 11 | 1−4T+pT2 |
| 13 | 1−2T+pT2 |
| 17 | 1−2T+pT2 |
| 19 | 1+4T+pT2 |
| 23 | 1+pT2 |
| 29 | 1−2T+pT2 |
| 31 | 1+pT2 |
| 37 | 1−10T+pT2 |
| 41 | 1−10T+pT2 |
| 43 | 1+4T+pT2 |
| 47 | 1−8T+pT2 |
| 53 | 1−10T+pT2 |
| 59 | 1−4T+pT2 |
| 61 | 1−2T+pT2 |
| 67 | 1+12T+pT2 |
| 71 | 1+8T+pT2 |
| 73 | 1−10T+pT2 |
| 79 | 1+pT2 |
| 83 | 1+12T+pT2 |
| 89 | 1+6T+pT2 |
| 97 | 1−2T+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
−9.907104671360377962786893632766, −9.080341213166513123570518780286, −8.438902886313495638742074033009, −7.60125180356652570547951985370, −6.67875901406960725991694880905, −5.85130665991850657402735978605, −4.38871739502786245271092399731, −3.82349173075266898382654912582, −2.63252540175087701435373844808, −1.18815390030287753587945796282,
1.18815390030287753587945796282, 2.63252540175087701435373844808, 3.82349173075266898382654912582, 4.38871739502786245271092399731, 5.85130665991850657402735978605, 6.67875901406960725991694880905, 7.60125180356652570547951985370, 8.438902886313495638742074033009, 9.080341213166513123570518780286, 9.907104671360377962786893632766