L(s) = 1 | + 2-s − 4-s − 5-s + 4·7-s − 3·8-s − 10-s − 4·11-s − 2·13-s + 4·14-s − 16-s − 2·17-s + 20-s − 4·22-s + 4·23-s + 25-s − 2·26-s − 4·28-s − 2·29-s + 5·32-s − 2·34-s − 4·35-s + 6·37-s + 3·40-s − 6·41-s + 8·43-s + 4·44-s + 4·46-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s − 1.06·8-s − 0.316·10-s − 1.20·11-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.485·17-s + 0.223·20-s − 0.852·22-s + 0.834·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s − 0.371·29-s + 0.883·32-s − 0.342·34-s − 0.676·35-s + 0.986·37-s + 0.474·40-s − 0.937·41-s + 1.21·43-s + 0.603·44-s + 0.589·46-s + ⋯ |
Λ(s)=(=(16245s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(16245s/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 | 3 | 1 |
| 5 | 1+T |
| 19 | 1 |
good | 2 | 1−T+pT2 |
| 7 | 1−4T+pT2 |
| 11 | 1+4T+pT2 |
| 13 | 1+2T+pT2 |
| 17 | 1+2T+pT2 |
| 23 | 1−4T+pT2 |
| 29 | 1+2T+pT2 |
| 31 | 1+pT2 |
| 37 | 1−6T+pT2 |
| 41 | 1+6T+pT2 |
| 43 | 1−8T+pT2 |
| 47 | 1−12T+pT2 |
| 53 | 1+14T+pT2 |
| 59 | 1−4T+pT2 |
| 61 | 1−14T+pT2 |
| 67 | 1−4T+pT2 |
| 71 | 1+pT2 |
| 73 | 1+14T+pT2 |
| 79 | 1+16T+pT2 |
| 83 | 1+pT2 |
| 89 | 1+6T+pT2 |
| 97 | 1−10T+pT2 |
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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
−15.87951060060536, −15.59866160048803, −14.92969482347150, −14.54083670355847, −14.14598147295844, −13.39652643823348, −12.93408284899613, −12.49489214274861, −11.72843046048931, −11.30169928554776, −10.78304187020159, −10.08298947583866, −9.344672989122720, −8.622783562343726, −8.276795980082519, −7.542233015618757, −7.150221255600763, −6.017915336794983, −5.420819903290018, −4.867253326391264, −4.504585766907321, −3.806809866912113, −2.825833181366516, −2.288620776094795, −1.081410149549704, 0,
1.081410149549704, 2.288620776094795, 2.825833181366516, 3.806809866912113, 4.504585766907321, 4.867253326391264, 5.420819903290018, 6.017915336794983, 7.150221255600763, 7.542233015618757, 8.276795980082519, 8.622783562343726, 9.344672989122720, 10.08298947583866, 10.78304187020159, 11.30169928554776, 11.72843046048931, 12.49489214274861, 12.93408284899613, 13.39652643823348, 14.14598147295844, 14.54083670355847, 14.92969482347150, 15.59866160048803, 15.87951060060536