L(s) = 1 | + 3·3-s + 4·5-s − 26·7-s + 9·9-s − 11·11-s + 32·13-s + 12·15-s + 74·17-s + 60·19-s − 78·21-s − 182·23-s − 109·25-s + 27·27-s + 90·29-s − 8·31-s − 33·33-s − 104·35-s + 66·37-s + 96·39-s + 422·41-s − 408·43-s + 36·45-s − 506·47-s + 333·49-s + 222·51-s − 348·53-s − 44·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.357·5-s − 1.40·7-s + 1/3·9-s − 0.301·11-s + 0.682·13-s + 0.206·15-s + 1.05·17-s + 0.724·19-s − 0.810·21-s − 1.64·23-s − 0.871·25-s + 0.192·27-s + 0.576·29-s − 0.0463·31-s − 0.174·33-s − 0.502·35-s + 0.293·37-s + 0.394·39-s + 1.60·41-s − 1.44·43-s + 0.119·45-s − 1.57·47-s + 0.970·49-s + 0.609·51-s − 0.901·53-s − 0.107·55-s + ⋯ |
Λ(s)=(=(2112s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(2112s/2ΓC(s+3/2)L(s)−Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−pT |
| 11 | 1+pT |
good | 5 | 1−4T+p3T2 |
| 7 | 1+26T+p3T2 |
| 13 | 1−32T+p3T2 |
| 17 | 1−74T+p3T2 |
| 19 | 1−60T+p3T2 |
| 23 | 1+182T+p3T2 |
| 29 | 1−90T+p3T2 |
| 31 | 1+8T+p3T2 |
| 37 | 1−66T+p3T2 |
| 41 | 1−422T+p3T2 |
| 43 | 1+408T+p3T2 |
| 47 | 1+506T+p3T2 |
| 53 | 1+348T+p3T2 |
| 59 | 1−200T+p3T2 |
| 61 | 1+132T+p3T2 |
| 67 | 1−1036T+p3T2 |
| 71 | 1−762T+p3T2 |
| 73 | 1+542T+p3T2 |
| 79 | 1+550T+p3T2 |
| 83 | 1−132T+p3T2 |
| 89 | 1−570T+p3T2 |
| 97 | 1−14T+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
−8.213967367680691300782723569916, −7.79449479861081698286700699396, −6.66022364838920303122687000699, −6.10248769674964699522562224397, −5.29682195345652693109702144393, −3.94927740677310572827003028734, −3.36199407088840525199809496499, −2.50193778729963704125352144638, −1.32505436075178096330294889103, 0,
1.32505436075178096330294889103, 2.50193778729963704125352144638, 3.36199407088840525199809496499, 3.94927740677310572827003028734, 5.29682195345652693109702144393, 6.10248769674964699522562224397, 6.66022364838920303122687000699, 7.79449479861081698286700699396, 8.213967367680691300782723569916