L(s) = 1 | − 0.561·5-s − 5.12·7-s − 11-s − 3.12·13-s − 2·17-s + 4·19-s − 6.56·23-s − 4.68·25-s + 3.12·29-s − 1.43·31-s + 2.87·35-s + 3.43·37-s − 7.12·41-s − 1.12·43-s − 8·47-s + 19.2·49-s − 4.24·53-s + 0.561·55-s − 12.8·59-s + 7.12·61-s + 1.75·65-s − 5.43·67-s − 3.68·71-s − 3.12·73-s + 5.12·77-s − 2.87·79-s + 9.12·83-s + ⋯ |
L(s) = 1 | − 0.251·5-s − 1.93·7-s − 0.301·11-s − 0.866·13-s − 0.485·17-s + 0.917·19-s − 1.36·23-s − 0.936·25-s + 0.579·29-s − 0.258·31-s + 0.486·35-s + 0.565·37-s − 1.11·41-s − 0.171·43-s − 1.16·47-s + 2.74·49-s − 0.583·53-s + 0.0757·55-s − 1.66·59-s + 0.912·61-s + 0.217·65-s − 0.664·67-s − 0.437·71-s − 0.365·73-s + 0.583·77-s − 0.323·79-s + 1.00·83-s + ⋯ |
Λ(s)=(=(6336s/2ΓC(s)L(s)Λ(2−s)
Λ(s)=(=(6336s/2ΓC(s+1/2)L(s)Λ(1−s)
Particular Values
L(1) |
≈ |
0.5159338435 |
L(21) |
≈ |
0.5159338435 |
L(23) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1 |
| 11 | 1+T |
good | 5 | 1+0.561T+5T2 |
| 7 | 1+5.12T+7T2 |
| 13 | 1+3.12T+13T2 |
| 17 | 1+2T+17T2 |
| 19 | 1−4T+19T2 |
| 23 | 1+6.56T+23T2 |
| 29 | 1−3.12T+29T2 |
| 31 | 1+1.43T+31T2 |
| 37 | 1−3.43T+37T2 |
| 41 | 1+7.12T+41T2 |
| 43 | 1+1.12T+43T2 |
| 47 | 1+8T+47T2 |
| 53 | 1+4.24T+53T2 |
| 59 | 1+12.8T+59T2 |
| 61 | 1−7.12T+61T2 |
| 67 | 1+5.43T+67T2 |
| 71 | 1+3.68T+71T2 |
| 73 | 1+3.12T+73T2 |
| 79 | 1+2.87T+79T2 |
| 83 | 1−9.12T+83T2 |
| 89 | 1−9.68T+89T2 |
| 97 | 1−11.4T+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
−7.84960030077535263021306549704, −7.38182180226491932793983993091, −6.49375406684251816548198980593, −6.12615412230479030103308315268, −5.22049160386946861150690571969, −4.32733388686746680024234025036, −3.46975669874966716870517664579, −2.93720590940063199864601979211, −1.97344929397874887580778724500, −0.34799344596824056453445742882,
0.34799344596824056453445742882, 1.97344929397874887580778724500, 2.93720590940063199864601979211, 3.46975669874966716870517664579, 4.32733388686746680024234025036, 5.22049160386946861150690571969, 6.12615412230479030103308315268, 6.49375406684251816548198980593, 7.38182180226491932793983993091, 7.84960030077535263021306549704