L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 2·5-s − 6·6-s + 8·8-s + 9·9-s − 4·10-s − 8·11-s − 12·12-s + 42·13-s + 6·15-s + 16·16-s + 2·17-s + 18·18-s + 124·19-s − 8·20-s − 16·22-s + 76·23-s − 24·24-s − 121·25-s + 84·26-s − 27·27-s + 254·29-s + 12·30-s + 72·31-s + 32·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.178·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.126·10-s − 0.219·11-s − 0.288·12-s + 0.896·13-s + 0.103·15-s + 1/4·16-s + 0.0285·17-s + 0.235·18-s + 1.49·19-s − 0.0894·20-s − 0.155·22-s + 0.689·23-s − 0.204·24-s − 0.967·25-s + 0.633·26-s − 0.192·27-s + 1.62·29-s + 0.0730·30-s + 0.417·31-s + 0.176·32-s + ⋯ |
Λ(s)=(=(294s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(294s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
2.444877403 |
L(21) |
≈ |
2.444877403 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−pT |
| 3 | 1+pT |
| 7 | 1 |
good | 5 | 1+2T+p3T2 |
| 11 | 1+8T+p3T2 |
| 13 | 1−42T+p3T2 |
| 17 | 1−2T+p3T2 |
| 19 | 1−124T+p3T2 |
| 23 | 1−76T+p3T2 |
| 29 | 1−254T+p3T2 |
| 31 | 1−72T+p3T2 |
| 37 | 1−398T+p3T2 |
| 41 | 1+462T+p3T2 |
| 43 | 1−212T+p3T2 |
| 47 | 1−264T+p3T2 |
| 53 | 1+162T+p3T2 |
| 59 | 1−772T+p3T2 |
| 61 | 1+30T+p3T2 |
| 67 | 1+764T+p3T2 |
| 71 | 1+236T+p3T2 |
| 73 | 1+418T+p3T2 |
| 79 | 1−552T+p3T2 |
| 83 | 1+1036T+p3T2 |
| 89 | 1+30T+p3T2 |
| 97 | 1−1190T+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
−11.56886698311922542653532608853, −10.63851879848623807711697193557, −9.673140973884217385418283947736, −8.299091446871014868611931534960, −7.24487164749901783641199687735, −6.20217852812527652318977227994, −5.30256143482064034171982614073, −4.20281487194666214110630342027, −2.94703327487831006010805444320, −1.09334361875456715075271536774,
1.09334361875456715075271536774, 2.94703327487831006010805444320, 4.20281487194666214110630342027, 5.30256143482064034171982614073, 6.20217852812527652318977227994, 7.24487164749901783641199687735, 8.299091446871014868611931534960, 9.673140973884217385418283947736, 10.63851879848623807711697193557, 11.56886698311922542653532608853