L(s) = 1 | − 2.08·2-s + 2.34·4-s − 0.992·5-s + 2.45·7-s − 0.726·8-s + 2.06·10-s − 0.745·11-s + 0.730·13-s − 5.11·14-s − 3.18·16-s + 4.76·17-s + 1.38·19-s − 2.33·20-s + 1.55·22-s − 5.31·23-s − 4.01·25-s − 1.52·26-s + 5.75·28-s − 0.563·31-s + 8.08·32-s − 9.93·34-s − 2.43·35-s + 9.49·37-s − 2.88·38-s + 0.721·40-s − 5.63·41-s + 11.0·43-s + ⋯ |
L(s) = 1 | − 1.47·2-s + 1.17·4-s − 0.443·5-s + 0.926·7-s − 0.256·8-s + 0.654·10-s − 0.224·11-s + 0.202·13-s − 1.36·14-s − 0.795·16-s + 1.15·17-s + 0.317·19-s − 0.521·20-s + 0.331·22-s − 1.10·23-s − 0.803·25-s − 0.298·26-s + 1.08·28-s − 0.101·31-s + 1.42·32-s − 1.70·34-s − 0.411·35-s + 1.56·37-s − 0.467·38-s + 0.114·40-s − 0.880·41-s + 1.67·43-s + ⋯ |
Λ(s)=(=(7569s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(7569s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.8696880939 |
L(21) |
≈ |
0.8696880939 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 29 | 1 |
good | 2 | 1+2.08T+2T2 |
| 5 | 1+0.992T+5T2 |
| 7 | 1−2.45T+7T2 |
| 11 | 1+0.745T+11T2 |
| 13 | 1−0.730T+13T2 |
| 17 | 1−4.76T+17T2 |
| 19 | 1−1.38T+19T2 |
| 23 | 1+5.31T+23T2 |
| 31 | 1+0.563T+31T2 |
| 37 | 1−9.49T+37T2 |
| 41 | 1+5.63T+41T2 |
| 43 | 1−11.0T+43T2 |
| 47 | 1+11.3T+47T2 |
| 53 | 1+4.53T+53T2 |
| 59 | 1+8.54T+59T2 |
| 61 | 1−8.56T+61T2 |
| 67 | 1−11.0T+67T2 |
| 71 | 1−6.10T+71T2 |
| 73 | 1−5.41T+73T2 |
| 79 | 1+7.67T+79T2 |
| 83 | 1−15.9T+83T2 |
| 89 | 1−9.90T+89T2 |
| 97 | 1−13.5T+97T2 |
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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
−7.943809846727732321762144843510, −7.74082513739303497250677872428, −6.76166742667370761071829255142, −5.94782559943401513647460228843, −5.10197189765695836699658790847, −4.30463536217657759912324231394, −3.44955369932712184545139997826, −2.29529959927873990136212201101, −1.52832090849494420631448839289, −0.61915879484559102634809750169,
0.61915879484559102634809750169, 1.52832090849494420631448839289, 2.29529959927873990136212201101, 3.44955369932712184545139997826, 4.30463536217657759912324231394, 5.10197189765695836699658790847, 5.94782559943401513647460228843, 6.76166742667370761071829255142, 7.74082513739303497250677872428, 7.943809846727732321762144843510