L(s) = 1 | + 2·3-s + 2·5-s + 7-s + 9-s − 11-s + 4·15-s + 4·17-s + 4·19-s + 2·21-s − 4·23-s − 25-s − 4·27-s + 2·29-s − 2·31-s − 2·33-s + 2·35-s − 6·37-s + 4·41-s − 4·43-s + 2·45-s + 2·47-s + 49-s + 8·51-s + 2·53-s − 2·55-s + 8·57-s − 6·59-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.338·35-s − 0.986·37-s + 0.624·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.269·55-s + 1.05·57-s − 0.781·59-s + ⋯ |
Λ(s)=(=(616s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(616s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.483500673 |
L(21) |
≈ |
2.483500673 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) | Isogeny Class over Fp |
---|
bad | 2 | 1 | |
| 7 | 1−T | |
| 11 | 1+T | |
good | 3 | 1−2T+pT2 | 1.3.ac |
| 5 | 1−2T+pT2 | 1.5.ac |
| 13 | 1+pT2 | 1.13.a |
| 17 | 1−4T+pT2 | 1.17.ae |
| 19 | 1−4T+pT2 | 1.19.ae |
| 23 | 1+4T+pT2 | 1.23.e |
| 29 | 1−2T+pT2 | 1.29.ac |
| 31 | 1+2T+pT2 | 1.31.c |
| 37 | 1+6T+pT2 | 1.37.g |
| 41 | 1−4T+pT2 | 1.41.ae |
| 43 | 1+4T+pT2 | 1.43.e |
| 47 | 1−2T+pT2 | 1.47.ac |
| 53 | 1−2T+pT2 | 1.53.ac |
| 59 | 1+6T+pT2 | 1.59.g |
| 61 | 1−4T+pT2 | 1.61.ae |
| 67 | 1+pT2 | 1.67.a |
| 71 | 1+12T+pT2 | 1.71.m |
| 73 | 1−16T+pT2 | 1.73.aq |
| 79 | 1+8T+pT2 | 1.79.i |
| 83 | 1+12T+pT2 | 1.83.m |
| 89 | 1−10T+pT2 | 1.89.ak |
| 97 | 1+2T+pT2 | 1.97.c |
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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
−10.28981135359909303761462495429, −9.721026942562876571917863787048, −8.900267729944967573194117141070, −8.051352337308495880478764667111, −7.35767416332621097285914793198, −5.98413065916960487951751915735, −5.19667268136125807114507816372, −3.75056949088258166882410190362, −2.72800026016992914882879677292, −1.66644659770392612638218462013,
1.66644659770392612638218462013, 2.72800026016992914882879677292, 3.75056949088258166882410190362, 5.19667268136125807114507816372, 5.98413065916960487951751915735, 7.35767416332621097285914793198, 8.051352337308495880478764667111, 8.900267729944967573194117141070, 9.721026942562876571917863787048, 10.28981135359909303761462495429