L(s) = 1 | + (2.25 + 1.30i)3-s + (−3.59 − 6.23i)5-s + (−5.36 − 4.49i)7-s + (−1.09 − 1.90i)9-s + (−3.81 − 2.20i)11-s − 18.1·13-s − 18.7i·15-s + (−8.58 + 14.8i)17-s + (19.4 − 11.2i)19-s + (−6.26 − 17.1i)21-s + (9.53 − 5.50i)23-s + (−13.4 + 23.2i)25-s − 29.2i·27-s + 37.4·29-s + (40.5 + 23.4i)31-s + ⋯ |
L(s) = 1 | + (0.752 + 0.434i)3-s + (−0.719 − 1.24i)5-s + (−0.766 − 0.641i)7-s + (−0.122 − 0.211i)9-s + (−0.347 − 0.200i)11-s − 1.39·13-s − 1.25i·15-s + (−0.504 + 0.874i)17-s + (1.02 − 0.590i)19-s + (−0.298 − 0.816i)21-s + (0.414 − 0.239i)23-s + (−0.536 + 0.929i)25-s − 1.08i·27-s + 1.29·29-s + (1.30 + 0.755i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.477 + 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.527434 - 0.887402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.527434 - 0.887402i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (5.36 + 4.49i)T \) |
good | 3 | \( 1 + (-2.25 - 1.30i)T + (4.5 + 7.79i)T^{2} \) |
| 5 | \( 1 + (3.59 + 6.23i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (3.81 + 2.20i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 18.1T + 169T^{2} \) |
| 17 | \( 1 + (8.58 - 14.8i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-19.4 + 11.2i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-9.53 + 5.50i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 37.4T + 841T^{2} \) |
| 31 | \( 1 + (-40.5 - 23.4i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (31.3 + 54.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 2.97T + 1.68e3T^{2} \) |
| 43 | \( 1 + 43.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (18.9 - 10.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-2.03 + 3.52i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (12.2 + 7.09i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (40.4 + 70.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-20.4 - 11.8i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 96.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-16.3 + 28.3i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-73.2 + 42.3i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 41.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (49.9 + 86.4i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 61.8T + 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.01819452498028187517037681656, −10.53402642215267700846117522621, −9.575487835020672540190510804016, −8.801854308769873053203948477999, −7.944555373004501288187392811419, −6.77705360934455112122725690242, −5.06085877783859191873564576392, −4.11390019977588875200821331188, −2.93222160767252822877972983254, −0.49436543176195877492833863613,
2.63455237363815592576812552998, 3.05811686380007414422785036111, 4.95830314758217337041006253893, 6.57953301941190128351121548951, 7.38114023352156044404912278225, 8.151939488612559405885050605841, 9.483889409627573871471098820074, 10.28862529298409062909931943144, 11.58921303138805957801877338740, 12.20106020289369793794024505867