L(s) = 1 | − 3-s − 26·7-s − 26·9-s − 45·11-s + 44·13-s + 117·17-s + 91·19-s + 26·21-s + 18·23-s + 53·27-s + 144·29-s − 26·31-s + 45·33-s − 214·37-s − 44·39-s − 459·41-s + 460·43-s + 468·47-s + 333·49-s − 117·51-s + 558·53-s − 91·57-s + 72·59-s − 118·61-s + 676·63-s − 251·67-s − 18·69-s + ⋯ |
L(s) = 1 | − 0.192·3-s − 1.40·7-s − 0.962·9-s − 1.23·11-s + 0.938·13-s + 1.66·17-s + 1.09·19-s + 0.270·21-s + 0.163·23-s + 0.377·27-s + 0.922·29-s − 0.150·31-s + 0.237·33-s − 0.950·37-s − 0.180·39-s − 1.74·41-s + 1.63·43-s + 1.45·47-s + 0.970·49-s − 0.321·51-s + 1.44·53-s − 0.211·57-s + 0.158·59-s − 0.247·61-s + 1.35·63-s − 0.457·67-s − 0.0314·69-s + ⋯ |
Λ(s)=(=(400s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(400s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
1.205452861 |
L(21) |
≈ |
1.205452861 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1 |
good | 3 | 1+T+p3T2 |
| 7 | 1+26T+p3T2 |
| 11 | 1+45T+p3T2 |
| 13 | 1−44T+p3T2 |
| 17 | 1−117T+p3T2 |
| 19 | 1−91T+p3T2 |
| 23 | 1−18T+p3T2 |
| 29 | 1−144T+p3T2 |
| 31 | 1+26T+p3T2 |
| 37 | 1+214T+p3T2 |
| 41 | 1+459T+p3T2 |
| 43 | 1−460T+p3T2 |
| 47 | 1−468T+p3T2 |
| 53 | 1−558T+p3T2 |
| 59 | 1−72T+p3T2 |
| 61 | 1+118T+p3T2 |
| 67 | 1+251T+p3T2 |
| 71 | 1+108T+p3T2 |
| 73 | 1−299T+p3T2 |
| 79 | 1−898T+p3T2 |
| 83 | 1+927T+p3T2 |
| 89 | 1−351T+p3T2 |
| 97 | 1−386T+p3T2 |
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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
−10.66222305581732317797491050676, −10.08665060484949798492308686817, −9.058462953622159937205941410219, −8.100774325401966563103217330172, −7.06472333355930260135293296809, −5.87965745953089203269089337582, −5.36091428648694381763533521097, −3.50693435780534881528933857626, −2.84020271325597463650178054580, −0.70634813288964357001921596267,
0.70634813288964357001921596267, 2.84020271325597463650178054580, 3.50693435780534881528933857626, 5.36091428648694381763533521097, 5.87965745953089203269089337582, 7.06472333355930260135293296809, 8.100774325401966563103217330172, 9.058462953622159937205941410219, 10.08665060484949798492308686817, 10.66222305581732317797491050676