L(s) = 1 | + 2-s − 1.84·3-s + 4-s − 5-s − 1.84·6-s − 5.01·7-s + 8-s + 0.414·9-s − 10-s − 4.69·11-s − 1.84·12-s + 3.83·13-s − 5.01·14-s + 1.84·15-s + 16-s − 5.78·17-s + 0.414·18-s + 5.19·19-s − 20-s + 9.25·21-s − 4.69·22-s + 6.40·23-s − 1.84·24-s + 25-s + 3.83·26-s + 4.77·27-s − 5.01·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.06·3-s + 0.5·4-s − 0.447·5-s − 0.754·6-s − 1.89·7-s + 0.353·8-s + 0.138·9-s − 0.316·10-s − 1.41·11-s − 0.533·12-s + 1.06·13-s − 1.33·14-s + 0.477·15-s + 0.250·16-s − 1.40·17-s + 0.0976·18-s + 1.19·19-s − 0.223·20-s + 2.02·21-s − 1.00·22-s + 1.33·23-s − 0.377·24-s + 0.200·25-s + 0.752·26-s + 0.919·27-s − 0.946·28-s + ⋯ |
Λ(s)=(=(9610s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(9610s/2ΓC(s+1/2)L(s)−Λ(1−s)
Particular Values
L(1) |
= |
0 |
L(21) |
= |
0 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−T |
| 5 | 1+T |
| 31 | 1 |
good | 3 | 1+1.84T+3T2 |
| 7 | 1+5.01T+7T2 |
| 11 | 1+4.69T+11T2 |
| 13 | 1−3.83T+13T2 |
| 17 | 1+5.78T+17T2 |
| 19 | 1−5.19T+19T2 |
| 23 | 1−6.40T+23T2 |
| 29 | 1−5.01T+29T2 |
| 37 | 1−1.86T+37T2 |
| 41 | 1+7.74T+41T2 |
| 43 | 1−0.904T+43T2 |
| 47 | 1−0.257T+47T2 |
| 53 | 1+3.83T+53T2 |
| 59 | 1−4.16T+59T2 |
| 61 | 1+2.99T+61T2 |
| 67 | 1+1.92T+67T2 |
| 71 | 1+5.17T+71T2 |
| 73 | 1+1.17T+73T2 |
| 79 | 1−14.8T+79T2 |
| 83 | 1−14.9T+83T2 |
| 89 | 1−13.5T+89T2 |
| 97 | 1+8.36T+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
−6.94537774491825474321855022161, −6.53398540560926640410736908268, −6.04360775327742847174187576290, −5.24068476672548678858500091700, −4.81688824701845601627966015169, −3.75064038521084321737536799683, −3.10447262354289391171114300363, −2.58824464373040444120139441430, −0.903409693238353169384284592919, 0,
0.903409693238353169384284592919, 2.58824464373040444120139441430, 3.10447262354289391171114300363, 3.75064038521084321737536799683, 4.81688824701845601627966015169, 5.24068476672548678858500091700, 6.04360775327742847174187576290, 6.53398540560926640410736908268, 6.94537774491825474321855022161