L(s) = 1 | + 1.65·2-s + 0.726·4-s − 0.377·7-s − 2.10·8-s + 1.37·11-s + 2.82·13-s − 0.622·14-s − 4.92·16-s − 6.37·17-s + 19-s + 2.27·22-s − 6.19·23-s + 4.65·26-s − 0.273·28-s + 3.37·29-s + 2.48·31-s − 3.92·32-s − 10.5·34-s + 5.58·37-s + 1.65·38-s − 8.50·41-s − 12.1·43-s + 1.00·44-s − 10.2·46-s + 6.87·47-s − 6.85·49-s + 2.04·52-s + ⋯ |
L(s) = 1 | + 1.16·2-s + 0.363·4-s − 0.142·7-s − 0.743·8-s + 0.415·11-s + 0.782·13-s − 0.166·14-s − 1.23·16-s − 1.54·17-s + 0.229·19-s + 0.484·22-s − 1.29·23-s + 0.913·26-s − 0.0517·28-s + 0.627·29-s + 0.445·31-s − 0.693·32-s − 1.80·34-s + 0.917·37-s + 0.267·38-s − 1.32·41-s − 1.85·43-s + 0.150·44-s − 1.50·46-s + 1.00·47-s − 0.979·49-s + 0.284·52-s + ⋯ |
Λ(s)=(=(4275s/2ΓC(s)L(s)−Λ(2−s)
Λ(s)=(=(4275s/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 |
| 19 | 1−T |
good | 2 | 1−1.65T+2T2 |
| 7 | 1+0.377T+7T2 |
| 11 | 1−1.37T+11T2 |
| 13 | 1−2.82T+13T2 |
| 17 | 1+6.37T+17T2 |
| 23 | 1+6.19T+23T2 |
| 29 | 1−3.37T+29T2 |
| 31 | 1−2.48T+31T2 |
| 37 | 1−5.58T+37T2 |
| 41 | 1+8.50T+41T2 |
| 43 | 1+12.1T+43T2 |
| 47 | 1−6.87T+47T2 |
| 53 | 1+11.5T+53T2 |
| 59 | 1+6.05T+59T2 |
| 61 | 1−5.02T+61T2 |
| 67 | 1−3.22T+67T2 |
| 71 | 1−2.30T+71T2 |
| 73 | 1+3.19T+73T2 |
| 79 | 1+6.71T+79T2 |
| 83 | 1+18.2T+83T2 |
| 89 | 1+1.50T+89T2 |
| 97 | 1+11.7T+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
−8.182234861577908092159455855413, −6.87611681630669854246862555928, −6.41430848517713743336462074270, −5.82639603082197173989983162781, −4.83934229253721202854902250315, −4.29757330439102968084705968159, −3.56618856496280352079943848112, −2.74835090251108993470981599065, −1.65945841947794841284978273456, 0,
1.65945841947794841284978273456, 2.74835090251108993470981599065, 3.56618856496280352079943848112, 4.29757330439102968084705968159, 4.83934229253721202854902250315, 5.82639603082197173989983162781, 6.41430848517713743336462074270, 6.87611681630669854246862555928, 8.182234861577908092159455855413