L(s) = 1 | + 4·2-s − 9·3-s + 16·4-s − 75.4·5-s − 36·6-s + 64·8-s + 81·9-s − 301.·10-s − 149.·11-s − 144·12-s + 349.·13-s + 679.·15-s + 256·16-s − 1.14e3·17-s + 324·18-s − 2.79e3·19-s − 1.20e3·20-s − 597.·22-s + 1.81e3·23-s − 576·24-s + 2.57e3·25-s + 1.39e3·26-s − 729·27-s − 759.·29-s + 2.71e3·30-s + 9.03e3·31-s + 1.02e3·32-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.35·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 0.954·10-s − 0.372·11-s − 0.288·12-s + 0.573·13-s + 0.779·15-s + 0.250·16-s − 0.964·17-s + 0.235·18-s − 1.77·19-s − 0.675·20-s − 0.263·22-s + 0.715·23-s − 0.204·24-s + 0.823·25-s + 0.405·26-s − 0.192·27-s − 0.167·29-s + 0.551·30-s + 1.68·31-s + 0.176·32-s + ⋯ |
Λ(s)=(=(294s/2ΓC(s)L(s)Λ(6−s)
Λ(s)=(=(294s/2ΓC(s+5/2)L(s)Λ(1−s)
Particular Values
L(3) |
≈ |
1.650518134 |
L(21) |
≈ |
1.650518134 |
L(27) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−4T |
| 3 | 1+9T |
| 7 | 1 |
good | 5 | 1+75.4T+3.12e3T2 |
| 11 | 1+149.T+1.61e5T2 |
| 13 | 1−349.T+3.71e5T2 |
| 17 | 1+1.14e3T+1.41e6T2 |
| 19 | 1+2.79e3T+2.47e6T2 |
| 23 | 1−1.81e3T+6.43e6T2 |
| 29 | 1+759.T+2.05e7T2 |
| 31 | 1−9.03e3T+2.86e7T2 |
| 37 | 1−7.79e3T+6.93e7T2 |
| 41 | 1−7.64e3T+1.15e8T2 |
| 43 | 1−1.21e4T+1.47e8T2 |
| 47 | 1−2.45e4T+2.29e8T2 |
| 53 | 1−1.35e4T+4.18e8T2 |
| 59 | 1+2.63e4T+7.14e8T2 |
| 61 | 1−3.53e4T+8.44e8T2 |
| 67 | 1−5.43e4T+1.35e9T2 |
| 71 | 1+7.01e4T+1.80e9T2 |
| 73 | 1+4.44e4T+2.07e9T2 |
| 79 | 1−6.16e4T+3.07e9T2 |
| 83 | 1+8.71e4T+3.93e9T2 |
| 89 | 1−9.85e4T+5.58e9T2 |
| 97 | 1−3.23e4T+8.58e9T2 |
show more | |
show less | |
L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−11.08979800950461414429127788226, −10.52083749087863782083296939466, −8.835175136012454143515272238903, −7.892436591561497192771935874752, −6.86741096192697068231315386162, −5.96893771827147647389312265714, −4.54192417118133853200706088505, −4.04388929474996239999347815924, −2.54027532383503418490992894949, −0.65932737831927643651496013680,
0.65932737831927643651496013680, 2.54027532383503418490992894949, 4.04388929474996239999347815924, 4.54192417118133853200706088505, 5.96893771827147647389312265714, 6.86741096192697068231315386162, 7.892436591561497192771935874752, 8.835175136012454143515272238903, 10.52083749087863782083296939466, 11.08979800950461414429127788226