L(s) = 1 | + 3·3-s + 4·5-s + 7·7-s + 9·9-s + 26·11-s + 2·13-s + 12·15-s − 36·17-s + 76·19-s + 21·21-s + 114·23-s − 109·25-s + 27·27-s + 6·29-s + 256·31-s + 78·33-s + 28·35-s − 86·37-s + 6·39-s + 160·41-s + 220·43-s + 36·45-s − 308·47-s + 49·49-s − 108·51-s + 258·53-s + 104·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.357·5-s + 0.377·7-s + 1/3·9-s + 0.712·11-s + 0.0426·13-s + 0.206·15-s − 0.513·17-s + 0.917·19-s + 0.218·21-s + 1.03·23-s − 0.871·25-s + 0.192·27-s + 0.0384·29-s + 1.48·31-s + 0.411·33-s + 0.135·35-s − 0.382·37-s + 0.0246·39-s + 0.609·41-s + 0.780·43-s + 0.119·45-s − 0.955·47-s + 1/7·49-s − 0.296·51-s + 0.668·53-s + 0.254·55-s + ⋯ |
Λ(s)=(=(336s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(336s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
2.729790786 |
L(21) |
≈ |
2.729790786 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1−pT |
| 7 | 1−pT |
good | 5 | 1−4T+p3T2 |
| 11 | 1−26T+p3T2 |
| 13 | 1−2T+p3T2 |
| 17 | 1+36T+p3T2 |
| 19 | 1−4pT+p3T2 |
| 23 | 1−114T+p3T2 |
| 29 | 1−6T+p3T2 |
| 31 | 1−256T+p3T2 |
| 37 | 1+86T+p3T2 |
| 41 | 1−160T+p3T2 |
| 43 | 1−220T+p3T2 |
| 47 | 1+308T+p3T2 |
| 53 | 1−258T+p3T2 |
| 59 | 1+264T+p3T2 |
| 61 | 1−606T+p3T2 |
| 67 | 1−520T+p3T2 |
| 71 | 1−286T+p3T2 |
| 73 | 1+530T+p3T2 |
| 79 | 1−44T+p3T2 |
| 83 | 1+1012T+p3T2 |
| 89 | 1−768T+p3T2 |
| 97 | 1−222T+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
−11.18202526156211682633030459213, −10.02116976844018075672574879923, −9.256754511391465422248791912843, −8.392643222239171722915915647728, −7.35761502852889108004468996777, −6.34926502809716269383707534680, −5.08227801250171614477862726395, −3.91284713877592745281501384708, −2.59326374861362459413048405266, −1.22145569389880188051329271277,
1.22145569389880188051329271277, 2.59326374861362459413048405266, 3.91284713877592745281501384708, 5.08227801250171614477862726395, 6.34926502809716269383707534680, 7.35761502852889108004468996777, 8.392643222239171722915915647728, 9.256754511391465422248791912843, 10.02116976844018075672574879923, 11.18202526156211682633030459213