L(s) = 1 | − 4.53·2-s + 12.5·4-s + 5·5-s − 20.5·8-s − 22.6·10-s + 19.0·11-s + 2.93·13-s − 7.21·16-s − 6.49·17-s + 5.43·19-s + 62.6·20-s − 86.3·22-s − 49.3·23-s + 25·25-s − 13.3·26-s + 291.·29-s − 244.·31-s + 196.·32-s + 29.4·34-s − 193.·37-s − 24.6·38-s − 102.·40-s + 315.·41-s − 300.·43-s + 238.·44-s + 223.·46-s + 86.5·47-s + ⋯ |
L(s) = 1 | − 1.60·2-s + 1.56·4-s + 0.447·5-s − 0.907·8-s − 0.716·10-s + 0.522·11-s + 0.0626·13-s − 0.112·16-s − 0.0927·17-s + 0.0656·19-s + 0.700·20-s − 0.837·22-s − 0.447·23-s + 0.200·25-s − 0.100·26-s + 1.86·29-s − 1.41·31-s + 1.08·32-s + 0.148·34-s − 0.858·37-s − 0.105·38-s − 0.405·40-s + 1.20·41-s − 1.06·43-s + 0.818·44-s + 0.717·46-s + 0.268·47-s + ⋯ |
Λ(s)=(=(2205s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(2205s/2ΓC(s+3/2)L(s)−Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 3 | 1 |
| 5 | 1−5T |
| 7 | 1 |
good | 2 | 1+4.53T+8T2 |
| 11 | 1−19.0T+1.33e3T2 |
| 13 | 1−2.93T+2.19e3T2 |
| 17 | 1+6.49T+4.91e3T2 |
| 19 | 1−5.43T+6.85e3T2 |
| 23 | 1+49.3T+1.21e4T2 |
| 29 | 1−291.T+2.43e4T2 |
| 31 | 1+244.T+2.97e4T2 |
| 37 | 1+193.T+5.06e4T2 |
| 41 | 1−315.T+6.89e4T2 |
| 43 | 1+300.T+7.95e4T2 |
| 47 | 1−86.5T+1.03e5T2 |
| 53 | 1+509.T+1.48e5T2 |
| 59 | 1+83.3T+2.05e5T2 |
| 61 | 1−5.25T+2.26e5T2 |
| 67 | 1−205.T+3.00e5T2 |
| 71 | 1+1.00e3T+3.57e5T2 |
| 73 | 1−1.00e3T+3.89e5T2 |
| 79 | 1+863.T+4.93e5T2 |
| 83 | 1−1.33e3T+5.71e5T2 |
| 89 | 1−326.T+7.04e5T2 |
| 97 | 1+1.52e3T+9.12e5T2 |
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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
−8.417399175437126460832182906878, −7.79367836251768885556992224732, −6.86169047475777195792345700455, −6.37518980802705233626552324238, −5.29522120612836792063810024495, −4.19960624411151701113767831248, −2.95031481132222464788993924340, −1.91716945934200517257247153403, −1.13811492539434948809810250831, 0,
1.13811492539434948809810250831, 1.91716945934200517257247153403, 2.95031481132222464788993924340, 4.19960624411151701113767831248, 5.29522120612836792063810024495, 6.37518980802705233626552324238, 6.86169047475777195792345700455, 7.79367836251768885556992224732, 8.417399175437126460832182906878