L(s) = 1 | + 0.523·2-s − 1.72·4-s + 3.20·7-s − 1.95·8-s − 5.20·11-s − 2·13-s + 1.67·14-s + 2.42·16-s − 17-s + 7.20·19-s − 2.72·22-s − 1.04·26-s − 5.52·28-s + 0.249·29-s + 6.40·31-s + 5.17·32-s − 0.523·34-s − 2.24·37-s + 3.77·38-s − 10.6·41-s − 4.09·43-s + 8.97·44-s − 11.7·47-s + 3.24·49-s + 3.45·52-s − 1.84·53-s − 6.24·56-s + ⋯ |
L(s) = 1 | + 0.370·2-s − 0.862·4-s + 1.21·7-s − 0.690·8-s − 1.56·11-s − 0.554·13-s + 0.448·14-s + 0.607·16-s − 0.242·17-s + 1.65·19-s − 0.581·22-s − 0.205·26-s − 1.04·28-s + 0.0463·29-s + 1.15·31-s + 0.915·32-s − 0.0898·34-s − 0.369·37-s + 0.612·38-s − 1.66·41-s − 0.624·43-s + 1.35·44-s − 1.70·47-s + 0.464·49-s + 0.478·52-s − 0.253·53-s − 0.835·56-s + ⋯ |
Λ(s)=(=(3825s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(3825s/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 |
| 17 | 1+T |
good | 2 | 1−0.523T+2T2 |
| 7 | 1−3.20T+7T2 |
| 11 | 1+5.20T+11T2 |
| 13 | 1+2T+13T2 |
| 19 | 1−7.20T+19T2 |
| 23 | 1+23T2 |
| 29 | 1−0.249T+29T2 |
| 31 | 1−6.40T+31T2 |
| 37 | 1+2.24T+37T2 |
| 41 | 1+10.6T+41T2 |
| 43 | 1+4.09T+43T2 |
| 47 | 1+11.7T+47T2 |
| 53 | 1+1.84T+53T2 |
| 59 | 1−10.4T+59T2 |
| 61 | 1+10.4T+61T2 |
| 67 | 1+10.3T+67T2 |
| 71 | 1−6.49T+71T2 |
| 73 | 1+3.84T+73T2 |
| 79 | 1−8T+79T2 |
| 83 | 1−8.80T+83T2 |
| 89 | 1+8.30T+89T2 |
| 97 | 1−4.49T+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
−8.044437080496471345196971314710, −7.69255023408653142314211767764, −6.62407693923615115514087328210, −5.42131649431182377395067522679, −5.07962152504064313832627078401, −4.62767744872691362034956928828, −3.43185779021311095998451336384, −2.66577050232533439714139098675, −1.41058966609962394730005058603, 0,
1.41058966609962394730005058603, 2.66577050232533439714139098675, 3.43185779021311095998451336384, 4.62767744872691362034956928828, 5.07962152504064313832627078401, 5.42131649431182377395067522679, 6.62407693923615115514087328210, 7.69255023408653142314211767764, 8.044437080496471345196971314710