L(s) = 1 | + 93·3-s + 125·5-s − 343·7-s + 6.46e3·9-s + 2.16e3·11-s − 1.66e3·13-s + 1.16e4·15-s − 3.57e4·17-s − 2.02e4·19-s − 3.18e4·21-s + 4.21e4·23-s + 1.56e4·25-s + 3.97e5·27-s − 1.11e5·29-s + 2.69e5·31-s + 2.01e5·33-s − 4.28e4·35-s + 5.32e5·37-s − 1.54e5·39-s + 1.58e5·41-s + 5.21e5·43-s + 8.07e5·45-s + 9.39e5·47-s + 1.17e5·49-s − 3.32e6·51-s − 4.08e5·53-s + 2.70e5·55-s + ⋯ |
L(s) = 1 | + 1.98·3-s + 0.447·5-s − 0.377·7-s + 2.95·9-s + 0.490·11-s − 0.209·13-s + 0.889·15-s − 1.76·17-s − 0.676·19-s − 0.751·21-s + 0.722·23-s + 1/5·25-s + 3.88·27-s − 0.851·29-s + 1.62·31-s + 0.976·33-s − 0.169·35-s + 1.72·37-s − 0.416·39-s + 0.358·41-s + 1.00·43-s + 1.32·45-s + 1.32·47-s + 1/7·49-s − 3.51·51-s − 0.376·53-s + 0.219·55-s + ⋯ |
Λ(s)=(=(560s/2ΓC(s)L(s)Λ(8−s)
Λ(s)=(=(560s/2ΓC(s+7/2)L(s)Λ(1−s)
Particular Values
L(4) |
≈ |
6.023461239 |
L(21) |
≈ |
6.023461239 |
L(29) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 5 | 1−p3T |
| 7 | 1+p3T |
good | 3 | 1−31pT+p7T2 |
| 11 | 1−197pT+p7T2 |
| 13 | 1+1661T+p7T2 |
| 17 | 1+35771T+p7T2 |
| 19 | 1+20222T+p7T2 |
| 23 | 1−42130T+p7T2 |
| 29 | 1+111789T+p7T2 |
| 31 | 1−269504T+p7T2 |
| 37 | 1−532774T+p7T2 |
| 41 | 1−158056T+p7T2 |
| 43 | 1−521874T+p7T2 |
| 47 | 1−939733T+p7T2 |
| 53 | 1+408384T+p7T2 |
| 59 | 1−522172T+p7T2 |
| 61 | 1−350080T+p7T2 |
| 67 | 1−3931176T+p7T2 |
| 71 | 1+1194016T+p7T2 |
| 73 | 1−998350T+p7T2 |
| 79 | 1−2120709T+p7T2 |
| 83 | 1−1746708T+p7T2 |
| 89 | 1+10077740T+p7T2 |
| 97 | 1+6238295T+p7T2 |
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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
−9.310856142700045964390073704600, −8.938784345521179340593204862736, −8.047947384715294617251081455186, −7.06989129710357733395268937244, −6.34828391652912194456009179766, −4.55782253145449557394573063654, −3.91989989836331594225487743865, −2.65823716486276100911316390675, −2.23930582823280443556232223492, −0.984664456895481641452636411850,
0.984664456895481641452636411850, 2.23930582823280443556232223492, 2.65823716486276100911316390675, 3.91989989836331594225487743865, 4.55782253145449557394573063654, 6.34828391652912194456009179766, 7.06989129710357733395268937244, 8.047947384715294617251081455186, 8.938784345521179340593204862736, 9.310856142700045964390073704600