L(s) = 1 | + 4·5-s − 7-s − 13-s − 8·17-s − 5·19-s + 4·23-s + 11·25-s − 4·29-s + 9·31-s − 4·35-s + 7·37-s + 8·41-s − 8·43-s + 8·47-s − 6·49-s − 8·53-s + 8·59-s + 7·61-s − 4·65-s + 67-s − 12·71-s + 5·73-s − 11·79-s − 12·83-s − 32·85-s − 8·89-s + 91-s + ⋯ |
L(s) = 1 | + 1.78·5-s − 0.377·7-s − 0.277·13-s − 1.94·17-s − 1.14·19-s + 0.834·23-s + 11/5·25-s − 0.742·29-s + 1.61·31-s − 0.676·35-s + 1.15·37-s + 1.24·41-s − 1.21·43-s + 1.16·47-s − 6/7·49-s − 1.09·53-s + 1.04·59-s + 0.896·61-s − 0.496·65-s + 0.122·67-s − 1.42·71-s + 0.585·73-s − 1.23·79-s − 1.31·83-s − 3.47·85-s − 0.847·89-s + 0.104·91-s + ⋯ |
Λ(s)=(=(226512s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(226512s/2ΓC(s+1/2)L(s)−Λ(1−s)
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 11 | 1 |
| 13 | 1+T |
good | 5 | 1−4T+pT2 |
| 7 | 1+T+pT2 |
| 17 | 1+8T+pT2 |
| 19 | 1+5T+pT2 |
| 23 | 1−4T+pT2 |
| 29 | 1+4T+pT2 |
| 31 | 1−9T+pT2 |
| 37 | 1−7T+pT2 |
| 41 | 1−8T+pT2 |
| 43 | 1+8T+pT2 |
| 47 | 1−8T+pT2 |
| 53 | 1+8T+pT2 |
| 59 | 1−8T+pT2 |
| 61 | 1−7T+pT2 |
| 67 | 1−T+pT2 |
| 71 | 1+12T+pT2 |
| 73 | 1−5T+pT2 |
| 79 | 1+11T+pT2 |
| 83 | 1+12T+pT2 |
| 89 | 1+8T+pT2 |
| 97 | 1−17T+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
−13.22902309742738, −12.93685816586909, −12.51800800291105, −11.66725032141753, −11.21716854067088, −10.82573515646371, −10.30955305785634, −9.876760194297676, −9.443313816021076, −9.093362888097233, −8.547899098554862, −8.225548569014377, −7.250363681221908, −6.834007306913862, −6.476276716999204, −5.974776767053898, −5.689321149212157, −4.785089996893599, −4.617971216725378, −3.996566490807287, −3.026444013604093, −2.558222533106761, −2.228675869903742, −1.617568777350902, −0.8820968748213705, 0,
0.8820968748213705, 1.617568777350902, 2.228675869903742, 2.558222533106761, 3.026444013604093, 3.996566490807287, 4.617971216725378, 4.785089996893599, 5.689321149212157, 5.974776767053898, 6.476276716999204, 6.834007306913862, 7.250363681221908, 8.225548569014377, 8.547899098554862, 9.093362888097233, 9.443313816021076, 9.876760194297676, 10.30955305785634, 10.82573515646371, 11.21716854067088, 11.66725032141753, 12.51800800291105, 12.93685816586909, 13.22902309742738