L(s) = 1 | + 26.5·2-s + 81·3-s + 190.·4-s + 1.29e3·5-s + 2.14e3·6-s − 3.38e3·7-s − 8.52e3·8-s + 6.56e3·9-s + 3.43e4·10-s + 8.29e4·11-s + 1.54e4·12-s + 1.56e5·13-s − 8.97e4·14-s + 1.05e5·15-s − 3.23e5·16-s + 1.22e5·17-s + 1.73e5·18-s − 1.30e5·19-s + 2.47e5·20-s − 2.74e5·21-s + 2.19e6·22-s + 1.72e6·23-s − 6.90e5·24-s − 2.71e5·25-s + 4.13e6·26-s + 5.31e5·27-s − 6.44e5·28-s + ⋯ |
L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.372·4-s + 0.928·5-s + 0.676·6-s − 0.532·7-s − 0.735·8-s + 0.333·9-s + 1.08·10-s + 1.70·11-s + 0.214·12-s + 1.51·13-s − 0.624·14-s + 0.535·15-s − 1.23·16-s + 0.355·17-s + 0.390·18-s − 0.229·19-s + 0.345·20-s − 0.307·21-s + 2.00·22-s + 1.28·23-s − 0.424·24-s − 0.138·25-s + 1.77·26-s + 0.192·27-s − 0.198·28-s + ⋯ |
Λ(s)=(=(57s/2ΓC(s)L(s)Λ(10−s)
Λ(s)=(=(57s/2ΓC(s+9/2)L(s)Λ(1−s)
Particular Values
L(5) |
≈ |
5.144575876 |
L(21) |
≈ |
5.144575876 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1−81T |
| 19 | 1+1.30e5T |
good | 2 | 1−26.5T+512T2 |
| 5 | 1−1.29e3T+1.95e6T2 |
| 7 | 1+3.38e3T+4.03e7T2 |
| 11 | 1−8.29e4T+2.35e9T2 |
| 13 | 1−1.56e5T+1.06e10T2 |
| 17 | 1−1.22e5T+1.18e11T2 |
| 23 | 1−1.72e6T+1.80e12T2 |
| 29 | 1+1.25e5T+1.45e13T2 |
| 31 | 1+2.93e6T+2.64e13T2 |
| 37 | 1−6.00e5T+1.29e14T2 |
| 41 | 1−2.80e7T+3.27e14T2 |
| 43 | 1+1.16e7T+5.02e14T2 |
| 47 | 1+8.08e6T+1.11e15T2 |
| 53 | 1+2.67e7T+3.29e15T2 |
| 59 | 1−9.71e7T+8.66e15T2 |
| 61 | 1+1.59e8T+1.16e16T2 |
| 67 | 1+2.71e8T+2.72e16T2 |
| 71 | 1−3.38e8T+4.58e16T2 |
| 73 | 1+7.79e7T+5.88e16T2 |
| 79 | 1+5.96e8T+1.19e17T2 |
| 83 | 1+7.10e7T+1.86e17T2 |
| 89 | 1+1.04e9T+3.50e17T2 |
| 97 | 1−3.58e8T+7.60e17T2 |
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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
−13.44147254156790786017096318296, −12.63519953212839916704914774145, −11.25375130056020728367951185724, −9.529606262605644189790777064501, −8.837884647736523485960891868210, −6.64361359829163123598492083350, −5.82301459918773624724813704423, −4.12659123642001577064196468572, −3.13822252147872819712721812270, −1.40469625205078531351679646561,
1.40469625205078531351679646561, 3.13822252147872819712721812270, 4.12659123642001577064196468572, 5.82301459918773624724813704423, 6.64361359829163123598492083350, 8.837884647736523485960891868210, 9.529606262605644189790777064501, 11.25375130056020728367951185724, 12.63519953212839916704914774145, 13.44147254156790786017096318296