L(s) = 1 | − 9·3-s + 34·5-s + 81·9-s − 332·11-s + 1.02e3·13-s − 306·15-s − 922·17-s − 452·19-s − 3.77e3·23-s − 1.96e3·25-s − 729·27-s + 1.16e3·29-s + 9.79e3·31-s + 2.98e3·33-s + 2.39e3·37-s − 9.23e3·39-s + 7.23e3·41-s + 4.65e3·43-s + 2.75e3·45-s − 2.46e4·47-s + 8.29e3·51-s + 1.11e3·53-s − 1.12e4·55-s + 4.06e3·57-s − 4.68e4·59-s + 9.76e3·61-s + 3.48e4·65-s + ⋯ |
L(s) = 1 | − 0.577·3-s + 0.608·5-s + 1/3·9-s − 0.827·11-s + 1.68·13-s − 0.351·15-s − 0.773·17-s − 0.287·19-s − 1.48·23-s − 0.630·25-s − 0.192·27-s + 0.257·29-s + 1.83·31-s + 0.477·33-s + 0.287·37-s − 0.972·39-s + 0.671·41-s + 0.383·43-s + 0.202·45-s − 1.62·47-s + 0.446·51-s + 0.0542·53-s − 0.503·55-s + 0.165·57-s − 1.75·59-s + 0.335·61-s + 1.02·65-s + ⋯ |
Λ(s)=(=(588s/2ΓC(s)L(s)−Λ(6−s)
Λ(s)=(=(588s/2ΓC(s+5/2)L(s)−Λ(1−s)
Particular Values
L(3) |
= |
0 |
L(21) |
= |
0 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1 |
| 3 | 1+p2T |
| 7 | 1 |
good | 5 | 1−34T+p5T2 |
| 11 | 1+332T+p5T2 |
| 13 | 1−1026T+p5T2 |
| 17 | 1+922T+p5T2 |
| 19 | 1+452T+p5T2 |
| 23 | 1+3776T+p5T2 |
| 29 | 1−1166T+p5T2 |
| 31 | 1−9792T+p5T2 |
| 37 | 1−2390T+p5T2 |
| 41 | 1−7230T+p5T2 |
| 43 | 1−4652T+p5T2 |
| 47 | 1+24672T+p5T2 |
| 53 | 1−1110T+p5T2 |
| 59 | 1+46892T+p5T2 |
| 61 | 1−9762T+p5T2 |
| 67 | 1+26252T+p5T2 |
| 71 | 1−65440T+p5T2 |
| 73 | 1−5606T+p5T2 |
| 79 | 1+9840T+p5T2 |
| 83 | 1+61108T+p5T2 |
| 89 | 1−62958T+p5T2 |
| 97 | 1−37838T+p5T2 |
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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
−9.657354100114722202553970356612, −8.526089349988505364416914856338, −7.80925921821835176361460696703, −6.23941364260332516456079401392, −6.17492297730751592398327003769, −4.88346513479350880461127393711, −3.87671114838761962616602096941, −2.45757359299401721087276887524, −1.32760186561171369487492882326, 0,
1.32760186561171369487492882326, 2.45757359299401721087276887524, 3.87671114838761962616602096941, 4.88346513479350880461127393711, 6.17492297730751592398327003769, 6.23941364260332516456079401392, 7.80925921821835176361460696703, 8.526089349988505364416914856338, 9.657354100114722202553970356612