L(s) = 1 | + 5·7-s + 7·13-s + 19-s − 5·25-s − 4·31-s + 37-s − 8·43-s + 18·49-s + 13·61-s − 11·67-s + 17·73-s − 13·79-s + 35·91-s + 5·97-s − 7·103-s − 2·109-s + ⋯ |
L(s) = 1 | + 1.88·7-s + 1.94·13-s + 0.229·19-s − 25-s − 0.718·31-s + 0.164·37-s − 1.21·43-s + 18/7·49-s + 1.66·61-s − 1.34·67-s + 1.98·73-s − 1.46·79-s + 3.66·91-s + 0.507·97-s − 0.689·103-s − 0.191·109-s + ⋯ |
Λ(s)=(=(1728s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(1728s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
2.360844297 |
L(21) |
≈ |
2.360844297 |
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 | |
| 3 | 1 | |
good | 5 | 1+pT2 | 1.5.a |
| 7 | 1−5T+pT2 | 1.7.af |
| 11 | 1+pT2 | 1.11.a |
| 13 | 1−7T+pT2 | 1.13.ah |
| 17 | 1+pT2 | 1.17.a |
| 19 | 1−T+pT2 | 1.19.ab |
| 23 | 1+pT2 | 1.23.a |
| 29 | 1+pT2 | 1.29.a |
| 31 | 1+4T+pT2 | 1.31.e |
| 37 | 1−T+pT2 | 1.37.ab |
| 41 | 1+pT2 | 1.41.a |
| 43 | 1+8T+pT2 | 1.43.i |
| 47 | 1+pT2 | 1.47.a |
| 53 | 1+pT2 | 1.53.a |
| 59 | 1+pT2 | 1.59.a |
| 61 | 1−13T+pT2 | 1.61.an |
| 67 | 1+11T+pT2 | 1.67.l |
| 71 | 1+pT2 | 1.71.a |
| 73 | 1−17T+pT2 | 1.73.ar |
| 79 | 1+13T+pT2 | 1.79.n |
| 83 | 1+pT2 | 1.83.a |
| 89 | 1+pT2 | 1.89.a |
| 97 | 1−5T+pT2 | 1.97.af |
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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
−9.125589838381877994041862400986, −8.360413074571016534853094652479, −8.001529503824122494641395863146, −7.02193571198558069600312241264, −5.94215330426161837619813679812, −5.28577670060288457890184502723, −4.31915400924746551561976811972, −3.53995560336243251785891738493, −2.01230440740899716446600775646, −1.20941721581339133701993176268,
1.20941721581339133701993176268, 2.01230440740899716446600775646, 3.53995560336243251785891738493, 4.31915400924746551561976811972, 5.28577670060288457890184502723, 5.94215330426161837619813679812, 7.02193571198558069600312241264, 8.001529503824122494641395863146, 8.360413074571016534853094652479, 9.125589838381877994041862400986