L(s) = 1 | + 522.·5-s − 265.·7-s − 4.30e3·11-s + 5.22e3·13-s + 9.35e3·17-s + 5.08e4·19-s − 5.48e4·23-s + 1.94e5·25-s + 1.78e5·29-s − 9.24e4·31-s − 1.38e5·35-s − 4.50e5·37-s + 4.01e5·41-s + 1.87e5·43-s − 3.87e5·47-s − 7.53e5·49-s + 1.05e6·53-s − 2.25e6·55-s − 7.03e5·59-s + 2.53e6·61-s + 2.72e6·65-s + 1.10e6·67-s + 3.82e6·71-s − 1.95e6·73-s + 1.14e6·77-s + 8.46e6·79-s + 1.93e6·83-s + ⋯ |
L(s) = 1 | + 1.86·5-s − 0.292·7-s − 0.976·11-s + 0.659·13-s + 0.461·17-s + 1.70·19-s − 0.939·23-s + 2.49·25-s + 1.35·29-s − 0.557·31-s − 0.546·35-s − 1.46·37-s + 0.908·41-s + 0.358·43-s − 0.544·47-s − 0.914·49-s + 0.971·53-s − 1.82·55-s − 0.445·59-s + 1.43·61-s + 1.23·65-s + 0.449·67-s + 1.26·71-s − 0.587·73-s + 0.285·77-s + 1.93·79-s + 0.372·83-s + ⋯ |
Λ(s)=(=(216s/2ΓC(s)L(s)Λ(8−s)
Λ(s)=(=(216s/2ΓC(s+7/2)L(s)Λ(1−s)
Particular Values
L(4) |
≈ |
3.181767678 |
L(21) |
≈ |
3.181767678 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
good | 5 | 1−522.T+7.81e4T2 |
| 7 | 1+265.T+8.23e5T2 |
| 11 | 1+4.30e3T+1.94e7T2 |
| 13 | 1−5.22e3T+6.27e7T2 |
| 17 | 1−9.35e3T+4.10e8T2 |
| 19 | 1−5.08e4T+8.93e8T2 |
| 23 | 1+5.48e4T+3.40e9T2 |
| 29 | 1−1.78e5T+1.72e10T2 |
| 31 | 1+9.24e4T+2.75e10T2 |
| 37 | 1+4.50e5T+9.49e10T2 |
| 41 | 1−4.01e5T+1.94e11T2 |
| 43 | 1−1.87e5T+2.71e11T2 |
| 47 | 1+3.87e5T+5.06e11T2 |
| 53 | 1−1.05e6T+1.17e12T2 |
| 59 | 1+7.03e5T+2.48e12T2 |
| 61 | 1−2.53e6T+3.14e12T2 |
| 67 | 1−1.10e6T+6.06e12T2 |
| 71 | 1−3.82e6T+9.09e12T2 |
| 73 | 1+1.95e6T+1.10e13T2 |
| 79 | 1−8.46e6T+1.92e13T2 |
| 83 | 1−1.93e6T+2.71e13T2 |
| 89 | 1+2.68e6T+4.42e13T2 |
| 97 | 1−4.35e6T+8.07e13T2 |
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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
−10.74171049194771486507493653240, −10.00627244303562048186991808883, −9.355145047736504987633839846075, −8.147405732479905932969936499301, −6.80151308716602436935356049926, −5.77104052499823986921611492930, −5.15297758630648953135467588763, −3.24854316226546024917919458372, −2.15479516993937576637358886634, −0.980957991107469459765747449631,
0.980957991107469459765747449631, 2.15479516993937576637358886634, 3.24854316226546024917919458372, 5.15297758630648953135467588763, 5.77104052499823986921611492930, 6.80151308716602436935356049926, 8.147405732479905932969936499301, 9.355145047736504987633839846075, 10.00627244303562048186991808883, 10.74171049194771486507493653240