L(s) = 1 | − 3.73·2-s + 5.94·4-s − 7·7-s + 7.67·8-s + 36.3·11-s + 75.8·13-s + 26.1·14-s − 76.2·16-s − 31.6·17-s + 25.1·19-s − 135.·22-s − 212.·23-s − 283.·26-s − 41.6·28-s + 235.·29-s − 270.·31-s + 223.·32-s + 118.·34-s − 362.·37-s − 93.9·38-s − 132.·41-s + 5.13·43-s + 216.·44-s + 792.·46-s − 216.·47-s + 49·49-s + 451.·52-s + ⋯ |
L(s) = 1 | − 1.32·2-s + 0.743·4-s − 0.377·7-s + 0.338·8-s + 0.996·11-s + 1.61·13-s + 0.499·14-s − 1.19·16-s − 0.451·17-s + 0.303·19-s − 1.31·22-s − 1.92·23-s − 2.13·26-s − 0.280·28-s + 1.50·29-s − 1.56·31-s + 1.23·32-s + 0.596·34-s − 1.60·37-s − 0.400·38-s − 0.505·41-s + 0.0182·43-s + 0.740·44-s + 2.54·46-s − 0.672·47-s + 0.142·49-s + 1.20·52-s + ⋯ |
Λ(s)=(=(1575s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(1575s/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 |
| 7 | 1+7T |
good | 2 | 1+3.73T+8T2 |
| 11 | 1−36.3T+1.33e3T2 |
| 13 | 1−75.8T+2.19e3T2 |
| 17 | 1+31.6T+4.91e3T2 |
| 19 | 1−25.1T+6.85e3T2 |
| 23 | 1+212.T+1.21e4T2 |
| 29 | 1−235.T+2.43e4T2 |
| 31 | 1+270.T+2.97e4T2 |
| 37 | 1+362.T+5.06e4T2 |
| 41 | 1+132.T+6.89e4T2 |
| 43 | 1−5.13T+7.95e4T2 |
| 47 | 1+216.T+1.03e5T2 |
| 53 | 1−455.T+1.48e5T2 |
| 59 | 1−689.T+2.05e5T2 |
| 61 | 1−130.T+2.26e5T2 |
| 67 | 1+633.T+3.00e5T2 |
| 71 | 1−1.06e3T+3.57e5T2 |
| 73 | 1+1.00e3T+3.89e5T2 |
| 79 | 1+381.T+4.93e5T2 |
| 83 | 1+48.5T+5.71e5T2 |
| 89 | 1+53.5T+7.04e5T2 |
| 97 | 1+968.T+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
−8.600482070559689503932093754458, −8.294818317983626520707626125187, −7.09742558722274511898808789670, −6.53927626910810476034512440344, −5.63046171702231332212903147999, −4.22798165998469826770686055857, −3.53593801720265316930473974944, −1.98062764744021305272744132256, −1.16416871364992294215086809502, 0,
1.16416871364992294215086809502, 1.98062764744021305272744132256, 3.53593801720265316930473974944, 4.22798165998469826770686055857, 5.63046171702231332212903147999, 6.53927626910810476034512440344, 7.09742558722274511898808789670, 8.294818317983626520707626125187, 8.600482070559689503932093754458