L(s) = 1 | + 0.347·2-s − 1.87·4-s + 0.652·5-s + 0.532·7-s − 1.34·8-s + 0.226·10-s − 11-s − 2.30·13-s + 0.184·14-s + 3.29·16-s − 4.41·17-s − 4.18·19-s − 1.22·20-s − 0.347·22-s + 1.41·23-s − 4.57·25-s − 0.800·26-s − 28-s + 6.35·29-s − 2.16·31-s + 3.83·32-s − 1.53·34-s + 0.347·35-s − 3.16·37-s − 1.45·38-s − 0.879·40-s − 9.31·41-s + ⋯ |
L(s) = 1 | + 0.245·2-s − 0.939·4-s + 0.291·5-s + 0.201·7-s − 0.476·8-s + 0.0716·10-s − 0.301·11-s − 0.639·13-s + 0.0493·14-s + 0.822·16-s − 1.06·17-s − 0.960·19-s − 0.274·20-s − 0.0740·22-s + 0.294·23-s − 0.914·25-s − 0.157·26-s − 0.188·28-s + 1.18·29-s − 0.388·31-s + 0.678·32-s − 0.262·34-s + 0.0587·35-s − 0.519·37-s − 0.235·38-s − 0.139·40-s − 1.45·41-s + ⋯ |
Λ(s)=(=(891s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(891s/2ΓC(s+1/2)L(s)−Λ(1−s)
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 11 | 1+T |
good | 2 | 1−0.347T+2T2 |
| 5 | 1−0.652T+5T2 |
| 7 | 1−0.532T+7T2 |
| 13 | 1+2.30T+13T2 |
| 17 | 1+4.41T+17T2 |
| 19 | 1+4.18T+19T2 |
| 23 | 1−1.41T+23T2 |
| 29 | 1−6.35T+29T2 |
| 31 | 1+2.16T+31T2 |
| 37 | 1+3.16T+37T2 |
| 41 | 1+9.31T+41T2 |
| 43 | 1+12.2T+43T2 |
| 47 | 1+3.86T+47T2 |
| 53 | 1−12.6T+53T2 |
| 59 | 1+3.23T+59T2 |
| 61 | 1+8.53T+61T2 |
| 67 | 1−4.96T+67T2 |
| 71 | 1−9.98T+71T2 |
| 73 | 1+7.49T+73T2 |
| 79 | 1+4.87T+79T2 |
| 83 | 1+4.59T+83T2 |
| 89 | 1+16.9T+89T2 |
| 97 | 1−12.6T+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
−9.813528084012545849001413704947, −8.715451400683312969934453774565, −8.322185163147674644761020573079, −7.06579729254783944256956272214, −6.14106593976144323564404634477, −5.05887063621192910141549236423, −4.50617322891500742157460201944, −3.31908492930294815809901345461, −1.97679894025303810451404739737, 0,
1.97679894025303810451404739737, 3.31908492930294815809901345461, 4.50617322891500742157460201944, 5.05887063621192910141549236423, 6.14106593976144323564404634477, 7.06579729254783944256956272214, 8.322185163147674644761020573079, 8.715451400683312969934453774565, 9.813528084012545849001413704947