L(s) = 1 | − 3-s + 2·7-s + 9-s − 4·11-s + 6·13-s + 6·17-s − 2·21-s + 4·23-s − 5·25-s − 27-s + 4·29-s + 10·31-s + 4·33-s + 2·37-s − 6·39-s − 2·41-s + 8·43-s − 12·47-s − 3·49-s − 6·51-s − 12·53-s − 4·59-s + 2·61-s + 2·63-s + 4·67-s − 4·69-s − 4·71-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s + 1.66·13-s + 1.45·17-s − 0.436·21-s + 0.834·23-s − 25-s − 0.192·27-s + 0.742·29-s + 1.79·31-s + 0.696·33-s + 0.328·37-s − 0.960·39-s − 0.312·41-s + 1.21·43-s − 1.75·47-s − 3/7·49-s − 0.840·51-s − 1.64·53-s − 0.520·59-s + 0.256·61-s + 0.251·63-s + 0.488·67-s − 0.481·69-s − 0.474·71-s + ⋯ |
Λ(s)=(=(384s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(384s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.265414252 |
L(21) |
≈ |
1.265414252 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+T |
good | 5 | 1+pT2 |
| 7 | 1−2T+pT2 |
| 11 | 1+4T+pT2 |
| 13 | 1−6T+pT2 |
| 17 | 1−6T+pT2 |
| 19 | 1+pT2 |
| 23 | 1−4T+pT2 |
| 29 | 1−4T+pT2 |
| 31 | 1−10T+pT2 |
| 37 | 1−2T+pT2 |
| 41 | 1+2T+pT2 |
| 43 | 1−8T+pT2 |
| 47 | 1+12T+pT2 |
| 53 | 1+12T+pT2 |
| 59 | 1+4T+pT2 |
| 61 | 1−2T+pT2 |
| 67 | 1−4T+pT2 |
| 71 | 1+4T+pT2 |
| 73 | 1+10T+pT2 |
| 79 | 1+6T+pT2 |
| 83 | 1−12T+pT2 |
| 89 | 1−2T+pT2 |
| 97 | 1+6T+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
−11.26026602445558485918915487570, −10.59394573181229341824784268538, −9.710853337608498914769336043468, −8.272861522300808832842531458007, −7.83756825226352882885863464519, −6.40139715881748782453956361829, −5.52928834758807112839354938677, −4.57667813854686016923499181308, −3.12662782741322079682173901720, −1.27109672072212401682622254955,
1.27109672072212401682622254955, 3.12662782741322079682173901720, 4.57667813854686016923499181308, 5.52928834758807112839354938677, 6.40139715881748782453956361829, 7.83756825226352882885863464519, 8.272861522300808832842531458007, 9.710853337608498914769336043468, 10.59394573181229341824784268538, 11.26026602445558485918915487570