L(s) = 1 | + 2·2-s + 4·4-s − 7-s + 8·8-s − 42·11-s − 67·13-s − 2·14-s + 16·16-s − 54·17-s − 115·19-s − 84·22-s + 162·23-s − 134·26-s − 4·28-s + 210·29-s − 193·31-s + 32·32-s − 108·34-s − 286·37-s − 230·38-s − 12·41-s + 263·43-s − 168·44-s + 324·46-s − 414·47-s − 342·49-s − 268·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.0539·7-s + 0.353·8-s − 1.15·11-s − 1.42·13-s − 0.0381·14-s + 1/4·16-s − 0.770·17-s − 1.38·19-s − 0.814·22-s + 1.46·23-s − 1.01·26-s − 0.0269·28-s + 1.34·29-s − 1.11·31-s + 0.176·32-s − 0.544·34-s − 1.27·37-s − 0.981·38-s − 0.0457·41-s + 0.932·43-s − 0.575·44-s + 1.03·46-s − 1.28·47-s − 0.997·49-s − 0.714·52-s + ⋯ |
Λ(s)=(=(450s/2ΓC(s)L(s)−Λ(4−s)
Λ(s)=(=(450s/2ΓC(s+3/2)L(s)−Λ(1−s)
Particular Values
L(2) |
= |
0 |
L(21) |
= |
0 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−pT |
| 3 | 1 |
| 5 | 1 |
good | 7 | 1+T+p3T2 |
| 11 | 1+42T+p3T2 |
| 13 | 1+67T+p3T2 |
| 17 | 1+54T+p3T2 |
| 19 | 1+115T+p3T2 |
| 23 | 1−162T+p3T2 |
| 29 | 1−210T+p3T2 |
| 31 | 1+193T+p3T2 |
| 37 | 1+286T+p3T2 |
| 41 | 1+12T+p3T2 |
| 43 | 1−263T+p3T2 |
| 47 | 1+414T+p3T2 |
| 53 | 1−192T+p3T2 |
| 59 | 1+690T+p3T2 |
| 61 | 1+733T+p3T2 |
| 67 | 1−299T+p3T2 |
| 71 | 1−228T+p3T2 |
| 73 | 1−938T+p3T2 |
| 79 | 1+160T+p3T2 |
| 83 | 1−462T+p3T2 |
| 89 | 1−240T+p3T2 |
| 97 | 1+511T+p3T2 |
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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
−10.53590891099361381586125100776, −9.383983358771029752170594587040, −8.286512841298656316969519409231, −7.27006923798158736534749385096, −6.47133082170900758326102848284, −5.15201860683402424611559286257, −4.59835388450073658273450453813, −3.07047455545715100279299436304, −2.12234604163918774211384388930, 0,
2.12234604163918774211384388930, 3.07047455545715100279299436304, 4.59835388450073658273450453813, 5.15201860683402424611559286257, 6.47133082170900758326102848284, 7.27006923798158736534749385096, 8.286512841298656316969519409231, 9.383983358771029752170594587040, 10.53590891099361381586125100776