L(s) = 1 | + 28·2-s − 116·3-s + 272·4-s + 625·5-s − 3.24e3·6-s + 2.40e3·7-s − 6.72e3·8-s − 6.22e3·9-s + 1.75e4·10-s − 2.55e4·11-s − 3.15e4·12-s − 4.23e4·13-s + 6.72e4·14-s − 7.25e4·15-s − 3.27e5·16-s − 5.26e5·17-s − 1.74e5·18-s − 3.50e5·19-s + 1.70e5·20-s − 2.78e5·21-s − 7.15e5·22-s − 6.21e5·23-s + 7.79e5·24-s + 3.90e5·25-s − 1.18e6·26-s + 3.00e6·27-s + 6.53e5·28-s + ⋯ |
L(s) = 1 | + 1.23·2-s − 0.826·3-s + 0.531·4-s + 0.447·5-s − 1.02·6-s + 0.377·7-s − 0.580·8-s − 0.316·9-s + 0.553·10-s − 0.526·11-s − 0.439·12-s − 0.410·13-s + 0.467·14-s − 0.369·15-s − 1.24·16-s − 1.52·17-s − 0.391·18-s − 0.616·19-s + 0.237·20-s − 0.312·21-s − 0.651·22-s − 0.463·23-s + 0.479·24-s + 1/5·25-s − 0.508·26-s + 1.08·27-s + 0.200·28-s + ⋯ |
Λ(s)=(=(35s/2ΓC(s)L(s)−Λ(10−s)
Λ(s)=(=(35s/2ΓC(s+9/2)L(s)−Λ(1−s)
Particular Values
L(5) |
= |
0 |
L(21) |
= |
0 |
L(211) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 5 | 1−p4T |
| 7 | 1−p4T |
good | 2 | 1−7p2T+p9T2 |
| 3 | 1+116T+p9T2 |
| 11 | 1+25548T+p9T2 |
| 13 | 1+42306T+p9T2 |
| 17 | 1+526342T+p9T2 |
| 19 | 1+350060T+p9T2 |
| 23 | 1+621976T+p9T2 |
| 29 | 1−6720430T+p9T2 |
| 31 | 1+6412208T+p9T2 |
| 37 | 1+2317682T+p9T2 |
| 41 | 1+10224678T+p9T2 |
| 43 | 1−30114004T+p9T2 |
| 47 | 1+23644912T+p9T2 |
| 53 | 1−57292654T+p9T2 |
| 59 | 1−84934780T+p9T2 |
| 61 | 1−14677822T+p9T2 |
| 67 | 1+244557812T+p9T2 |
| 71 | 1−61901952T+p9T2 |
| 73 | 1+283763726T+p9T2 |
| 79 | 1−276107480T+p9T2 |
| 83 | 1+72995956T+p9T2 |
| 89 | 1+896368470T+p9T2 |
| 97 | 1−1205809578T+p9T2 |
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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
−13.82706994463753367755988528351, −12.77780797516303524791171987966, −11.71522569981137308570261655711, −10.60093104081596524156837200727, −8.757475580146295307777216922596, −6.60214279484499050801103175578, −5.49379448429270748142859265401, −4.47552726515895085107292806570, −2.48792095993394324947306954592, 0,
2.48792095993394324947306954592, 4.47552726515895085107292806570, 5.49379448429270748142859265401, 6.60214279484499050801103175578, 8.757475580146295307777216922596, 10.60093104081596524156837200727, 11.71522569981137308570261655711, 12.77780797516303524791171987966, 13.82706994463753367755988528351