L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 3·11-s − 13-s + 2·14-s + 16-s − 8·17-s − 20-s − 3·22-s − 4·23-s − 4·25-s + 26-s − 2·28-s + 29-s − 3·31-s − 32-s + 8·34-s + 2·35-s + 8·37-s + 40-s − 2·41-s − 11·43-s + 3·44-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.904·11-s − 0.277·13-s + 0.534·14-s + 1/4·16-s − 1.94·17-s − 0.223·20-s − 0.639·22-s − 0.834·23-s − 4/5·25-s + 0.196·26-s − 0.377·28-s + 0.185·29-s − 0.538·31-s − 0.176·32-s + 1.37·34-s + 0.338·35-s + 1.31·37-s + 0.158·40-s − 0.312·41-s − 1.67·43-s + 0.452·44-s + ⋯ |
Λ(s)=(=(522s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(522s/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+T |
| 3 | 1 |
| 29 | 1−T |
good | 5 | 1+T+pT2 |
| 7 | 1+2T+pT2 |
| 11 | 1−3T+pT2 |
| 13 | 1+T+pT2 |
| 17 | 1+8T+pT2 |
| 19 | 1+pT2 |
| 23 | 1+4T+pT2 |
| 31 | 1+3T+pT2 |
| 37 | 1−8T+pT2 |
| 41 | 1+2T+pT2 |
| 43 | 1+11T+pT2 |
| 47 | 1+13T+pT2 |
| 53 | 1−11T+pT2 |
| 59 | 1+pT2 |
| 61 | 1+8T+pT2 |
| 67 | 1+12T+pT2 |
| 71 | 1+2T+pT2 |
| 73 | 1−4T+pT2 |
| 79 | 1−15T+pT2 |
| 83 | 1+4T+pT2 |
| 89 | 1−10T+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
−10.26866147802001152913025958268, −9.482258294877269881590757644436, −8.763894772751692239273117298832, −7.81012507517906630443430989651, −6.74819217569927040729981669784, −6.19818150900186047644659279635, −4.54621693456857955686454044680, −3.46305229707620473692544260201, −2.01064091456439163113006432735, 0,
2.01064091456439163113006432735, 3.46305229707620473692544260201, 4.54621693456857955686454044680, 6.19818150900186047644659279635, 6.74819217569927040729981669784, 7.81012507517906630443430989651, 8.763894772751692239273117298832, 9.482258294877269881590757644436, 10.26866147802001152913025958268