L(s) = 1 | + (1.5 − 0.866i)3-s + (0.5 − 2.59i)7-s + (1.5 − 2.59i)9-s + (3 − 1.73i)11-s + (−1.5 + 0.866i)13-s + (1.5 + 0.866i)17-s + (2.5 + 4.33i)19-s + (−1.5 − 4.33i)21-s + (−3 − 1.73i)23-s + (−2.5 − 4.33i)25-s − 5.19i·27-s + (−1.5 + 2.59i)29-s − 31-s + (3 − 5.19i)33-s + (−3.5 − 6.06i)37-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (0.188 − 0.981i)7-s + (0.5 − 0.866i)9-s + (0.904 − 0.522i)11-s + (−0.416 + 0.240i)13-s + (0.363 + 0.210i)17-s + (0.573 + 0.993i)19-s + (−0.327 − 0.944i)21-s + (−0.625 − 0.361i)23-s + (−0.5 − 0.866i)25-s − 0.999i·27-s + (−0.278 + 0.482i)29-s − 0.179·31-s + (0.522 − 0.904i)33-s + (−0.575 − 0.996i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.281 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.244068717\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.244068717\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 2.59i)T \) |
good | 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.5 - 0.866i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3 + 1.73i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.5 - 2.59i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.5 - 0.866i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.5 + 0.866i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9T + 47T^{2} \) |
| 53 | \( 1 + (-4.5 + 7.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 15T + 59T^{2} \) |
| 61 | \( 1 + 1.73iT - 61T^{2} \) |
| 67 | \( 1 - 15.5iT - 67T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-1.5 - 0.866i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 1.73iT - 79T^{2} \) |
| 83 | \( 1 + (-4.5 + 7.79i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.5 + 0.866i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.5 + 0.866i)T + (48.5 + 84.0i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.837147548740087360524857288925, −8.821488985283046628773388187527, −8.132590810976590708295117695620, −7.32063871918843431211814253998, −6.65791770779633399738343930165, −5.58985726301466298688621424048, −4.03601918940879559176678853399, −3.66367562909593689065816477613, −2.17742987794193203769958104679, −1.00566263904630957776191389759,
1.76862279691920699171063195345, 2.79519533687752134454813965518, 3.81101404945237639188635065524, 4.86595464522402419086456869466, 5.65013247888171610059278431792, 6.95969319067565421437770103243, 7.73371322268840453574308951633, 8.623244163171457247585221814413, 9.425171244275730096917465766313, 9.724511980708443519815528802470