L(s) = 1 | − 5-s − 0.113·7-s − 3.39·11-s + 6.27·13-s − 3.61·17-s − 1.39·19-s − 23-s + 25-s + 6.38·29-s + 9.45·31-s + 0.113·35-s − 7.78·37-s − 11.0·41-s + 1.55·43-s − 1.70·47-s − 6.98·49-s + 8.71·53-s + 3.39·55-s + 8.95·59-s − 1.83·61-s − 6.27·65-s + 9.21·67-s + 6.82·71-s − 2.27·73-s + 0.387·77-s + 11.7·79-s − 1.16·83-s + ⋯ |
L(s) = 1 | − 0.447·5-s − 0.0430·7-s − 1.02·11-s + 1.74·13-s − 0.877·17-s − 0.321·19-s − 0.208·23-s + 0.200·25-s + 1.18·29-s + 1.69·31-s + 0.0192·35-s − 1.28·37-s − 1.72·41-s + 0.237·43-s − 0.248·47-s − 0.998·49-s + 1.19·53-s + 0.458·55-s + 1.16·59-s − 0.234·61-s − 0.778·65-s + 1.12·67-s + 0.810·71-s − 0.266·73-s + 0.0441·77-s + 1.32·79-s − 0.127·83-s + ⋯ |
Λ(s)=(=(4140s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(4140s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
1.542007752 |
L(21) |
≈ |
1.542007752 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 5 | 1+T |
| 23 | 1+T |
good | 7 | 1+0.113T+7T2 |
| 11 | 1+3.39T+11T2 |
| 13 | 1−6.27T+13T2 |
| 17 | 1+3.61T+17T2 |
| 19 | 1+1.39T+19T2 |
| 29 | 1−6.38T+29T2 |
| 31 | 1−9.45T+31T2 |
| 37 | 1+7.78T+37T2 |
| 41 | 1+11.0T+41T2 |
| 43 | 1−1.55T+43T2 |
| 47 | 1+1.70T+47T2 |
| 53 | 1−8.71T+53T2 |
| 59 | 1−8.95T+59T2 |
| 61 | 1+1.83T+61T2 |
| 67 | 1−9.21T+67T2 |
| 71 | 1−6.82T+71T2 |
| 73 | 1+2.27T+73T2 |
| 79 | 1−11.7T+79T2 |
| 83 | 1+1.16T+83T2 |
| 89 | 1−1.77T+89T2 |
| 97 | 1+0.131T+97T2 |
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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
−8.453029916113806364068511724790, −7.913937187168915828813180784055, −6.72821316575325459541665485805, −6.44970009053304233000890405342, −5.38010373692381803314529641614, −4.65960319288485828245121282725, −3.79174508532790750144771235069, −3.02971922407385045009529678351, −1.98465952860745940031424126558, −0.70198113341966677257745759571,
0.70198113341966677257745759571, 1.98465952860745940031424126558, 3.02971922407385045009529678351, 3.79174508532790750144771235069, 4.65960319288485828245121282725, 5.38010373692381803314529641614, 6.44970009053304233000890405342, 6.72821316575325459541665485805, 7.913937187168915828813180784055, 8.453029916113806364068511724790