L(s) = 1 | + (−0.866 − 0.5i)2-s + (−2.41 − 1.39i)3-s + (0.499 + 0.866i)4-s + (1.39 + 2.41i)6-s − i·7-s − 0.999i·8-s + (2.39 + 4.14i)9-s − 3.79·11-s − 2.79i·12-s + (0.180 − 0.104i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (0.685 + 0.395i)17-s − 4.79i·18-s + (3.5 + 2.59i)19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (−1.39 − 0.805i)3-s + (0.249 + 0.433i)4-s + (0.569 + 0.986i)6-s − 0.377i·7-s − 0.353i·8-s + (0.798 + 1.38i)9-s − 1.14·11-s − 0.805i·12-s + (0.0501 − 0.0289i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.166 + 0.0959i)17-s − 1.12i·18-s + (0.802 + 0.596i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.884 + 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.513534 - 0.127211i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.513534 - 0.127211i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
good | 3 | \( 1 + (2.41 + 1.39i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + iT - 7T^{2} \) |
| 11 | \( 1 + 3.79T + 11T^{2} \) |
| 13 | \( 1 + (-0.180 + 0.104i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.685 - 0.395i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (3.96 - 2.29i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.39 - 5.88i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4.79T + 31T^{2} \) |
| 37 | \( 1 - 3.58iT - 37T^{2} \) |
| 41 | \( 1 + (-5.68 + 9.85i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.01 - 2.89i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (10.5 - 6.08i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.96 + 2.29i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.29 + 3.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.686 + 1.18i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-13.4 + 7.79i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.29 + 3.96i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.41 - 1.39i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.47 + 12.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.79iT - 83T^{2} \) |
| 89 | \( 1 + (-2.29 - 3.96i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.7 - 6.18i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.32427585032410138413663119845, −9.309078037807964879825895758169, −7.972177916026175980200020208402, −7.54894161546927194706364792636, −6.64734468879795903034504744964, −5.71654031731044196094814699168, −4.98198434722564228706857383841, −3.49466458255139254945401140441, −1.99394042394697031207350557019, −0.790790986160098835539034211647,
0.55359730061904770795446690490, 2.52908476112377962827003005296, 4.14680441371439956586378257520, 5.19309809017402361018799595982, 5.66418016906147588024101071534, 6.54557889587842084710198321524, 7.56797209476568396961214091199, 8.464132590591547869763579836148, 9.600099669855011967021450208115, 10.03219346756586581583451461785