L(s) = 1 | + 1.55·2-s − 4.96·3-s − 5.58·4-s − 7.71·6-s + 7·7-s − 21.1·8-s − 2.38·9-s + 29.8·11-s + 27.6·12-s + 90.7·13-s + 10.8·14-s + 11.7·16-s + 29.5·17-s − 3.70·18-s − 62.3·19-s − 34.7·21-s + 46.3·22-s + 90.6·23-s + 104.·24-s + 141.·26-s + 145.·27-s − 39.0·28-s + 193.·29-s − 152.·31-s + 187.·32-s − 147.·33-s + 45.9·34-s + ⋯ |
L(s) = 1 | + 0.549·2-s − 0.954·3-s − 0.697·4-s − 0.525·6-s + 0.377·7-s − 0.933·8-s − 0.0882·9-s + 0.817·11-s + 0.666·12-s + 1.93·13-s + 0.207·14-s + 0.184·16-s + 0.421·17-s − 0.0485·18-s − 0.752·19-s − 0.360·21-s + 0.449·22-s + 0.821·23-s + 0.891·24-s + 1.06·26-s + 1.03·27-s − 0.263·28-s + 1.23·29-s − 0.881·31-s + 1.03·32-s − 0.780·33-s + 0.232·34-s + ⋯ |
Λ(s)=(=(175s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(175s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
1.392043206 |
L(21) |
≈ |
1.392043206 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1 |
| 7 | 1−7T |
good | 2 | 1−1.55T+8T2 |
| 3 | 1+4.96T+27T2 |
| 11 | 1−29.8T+1.33e3T2 |
| 13 | 1−90.7T+2.19e3T2 |
| 17 | 1−29.5T+4.91e3T2 |
| 19 | 1+62.3T+6.85e3T2 |
| 23 | 1−90.6T+1.21e4T2 |
| 29 | 1−193.T+2.43e4T2 |
| 31 | 1+152.T+2.97e4T2 |
| 37 | 1−102.T+5.06e4T2 |
| 41 | 1+266.T+6.89e4T2 |
| 43 | 1−387.T+7.95e4T2 |
| 47 | 1+152.T+1.03e5T2 |
| 53 | 1+81.5T+1.48e5T2 |
| 59 | 1+235.T+2.05e5T2 |
| 61 | 1−510.T+2.26e5T2 |
| 67 | 1+347.T+3.00e5T2 |
| 71 | 1−317.T+3.57e5T2 |
| 73 | 1+709.T+3.89e5T2 |
| 79 | 1−1.06e3T+4.93e5T2 |
| 83 | 1−503.T+5.71e5T2 |
| 89 | 1−482.T+7.04e5T2 |
| 97 | 1−481.T+9.12e5T2 |
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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
−12.24564744287305817197683230721, −11.37700675714668244784993218792, −10.59071143359218619923739112422, −9.081497581070433486425702362394, −8.364339837620896382315771286701, −6.49656153616018966220551853942, −5.77818845190338978788656727815, −4.65521735189639904497631826090, −3.50941245521523224717411538051, −0.953443287762168989611222339201,
0.953443287762168989611222339201, 3.50941245521523224717411538051, 4.65521735189639904497631826090, 5.77818845190338978788656727815, 6.49656153616018966220551853942, 8.364339837620896382315771286701, 9.081497581070433486425702362394, 10.59071143359218619923739112422, 11.37700675714668244784993218792, 12.24564744287305817197683230721