L(s) = 1 | + 5-s − 4·7-s + 4·11-s + 8·17-s − 6·23-s + 25-s + 8·29-s − 2·31-s − 4·35-s + 4·37-s − 10·41-s + 4·43-s − 8·47-s + 9·49-s + 10·53-s + 4·55-s + 12·59-s − 2·61-s + 2·67-s − 12·71-s + 2·73-s − 16·77-s + 8·79-s − 8·83-s + 8·85-s − 6·89-s + 10·97-s + ⋯ |
L(s) = 1 | + 0.447·5-s − 1.51·7-s + 1.20·11-s + 1.94·17-s − 1.25·23-s + 1/5·25-s + 1.48·29-s − 0.359·31-s − 0.676·35-s + 0.657·37-s − 1.56·41-s + 0.609·43-s − 1.16·47-s + 9/7·49-s + 1.37·53-s + 0.539·55-s + 1.56·59-s − 0.256·61-s + 0.244·67-s − 1.42·71-s + 0.234·73-s − 1.82·77-s + 0.900·79-s − 0.878·83-s + 0.867·85-s − 0.635·89-s + 1.01·97-s + ⋯ |
Λ(s)=(=(60840s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(60840s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.506246097 |
L(21) |
≈ |
2.506246097 |
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 |
| 13 | 1 |
good | 7 | 1+4T+pT2 |
| 11 | 1−4T+pT2 |
| 17 | 1−8T+pT2 |
| 19 | 1+pT2 |
| 23 | 1+6T+pT2 |
| 29 | 1−8T+pT2 |
| 31 | 1+2T+pT2 |
| 37 | 1−4T+pT2 |
| 41 | 1+10T+pT2 |
| 43 | 1−4T+pT2 |
| 47 | 1+8T+pT2 |
| 53 | 1−10T+pT2 |
| 59 | 1−12T+pT2 |
| 61 | 1+2T+pT2 |
| 67 | 1−2T+pT2 |
| 71 | 1+12T+pT2 |
| 73 | 1−2T+pT2 |
| 79 | 1−8T+pT2 |
| 83 | 1+8T+pT2 |
| 89 | 1+6T+pT2 |
| 97 | 1−10T+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
−14.31827682268330, −13.75529959392353, −13.38843650532507, −12.65098837626722, −12.34659596848539, −11.84493725498160, −11.47329817351712, −10.42129166923081, −10.06022719034009, −9.824041301100555, −9.339548223380404, −8.598482661951749, −8.237875990252604, −7.359679844869202, −6.915472804319874, −6.326799811113852, −5.957173063839778, −5.462753876839657, −4.614963867912242, −3.867665009443891, −3.420918059312280, −2.929598945332072, −2.078079931708963, −1.253138520025687, −0.5833846364321820,
0.5833846364321820, 1.253138520025687, 2.078079931708963, 2.929598945332072, 3.420918059312280, 3.867665009443891, 4.614963867912242, 5.462753876839657, 5.957173063839778, 6.326799811113852, 6.915472804319874, 7.359679844869202, 8.237875990252604, 8.598482661951749, 9.339548223380404, 9.824041301100555, 10.06022719034009, 10.42129166923081, 11.47329817351712, 11.84493725498160, 12.34659596848539, 12.65098837626722, 13.38843650532507, 13.75529959392353, 14.31827682268330