L(s) = 1 | + (−0.984 − 0.173i)2-s + (−0.223 + 0.266i)3-s + (0.939 + 0.342i)4-s + (0.266 − 0.223i)6-s + (1.52 − 0.879i)7-s + (−0.866 − 0.5i)8-s + (0.5 + 2.83i)9-s + (−2.11 + 3.66i)11-s + (−0.300 + 0.173i)12-s + (0.684 + 0.815i)13-s + (−1.65 + 0.601i)14-s + (0.766 + 0.642i)16-s + (−7.02 − 1.23i)17-s − 2.87i·18-s + (−3.93 + 1.86i)19-s + ⋯ |
L(s) = 1 | + (−0.696 − 0.122i)2-s + (−0.128 + 0.153i)3-s + (0.469 + 0.171i)4-s + (0.108 − 0.0911i)6-s + (0.575 − 0.332i)7-s + (−0.306 − 0.176i)8-s + (0.166 + 0.945i)9-s + (−0.637 + 1.10i)11-s + (−0.0868 + 0.0501i)12-s + (0.189 + 0.226i)13-s + (−0.441 + 0.160i)14-s + (0.191 + 0.160i)16-s + (−1.70 − 0.300i)17-s − 0.678i·18-s + (−0.903 + 0.427i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.241980 + 0.530900i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.241980 + 0.530900i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 + 0.173i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (3.93 - 1.86i)T \) |
good | 3 | \( 1 + (0.223 - 0.266i)T + (-0.520 - 2.95i)T^{2} \) |
| 7 | \( 1 + (-1.52 + 0.879i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.11 - 3.66i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.684 - 0.815i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (7.02 + 1.23i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-1.28 + 3.53i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.10 + 6.27i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.41 - 7.64i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.45iT - 37T^{2} \) |
| 41 | \( 1 + (-1.43 - 1.20i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (1.26 + 3.47i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (3.61 - 0.638i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (3.38 - 9.29i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.26 + 7.18i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.98 + 1.81i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (11.4 - 2.02i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.65 - 0.965i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (0.509 - 0.607i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-5.12 - 4.30i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.30 + 0.754i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (9.12 - 7.65i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.85 + 0.326i)T + (91.1 + 33.1i)T^{2} \) |
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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
−10.49285270214410466634342266199, −9.612077440484194876842863623231, −8.571961362592408493471613346450, −7.976267211431749202387891382789, −7.10708335541646761620752093305, −6.30148726968317811564385958815, −4.76797803254561519395890244066, −4.44524960912609691341389611257, −2.58418152274852687062925810556, −1.73873978338435342220101982829,
0.32999115519864626917321960981, 1.87448446011748354493572469758, 3.12134264800550263369224527904, 4.42061390434052226020897330701, 5.66450960721122756571791105369, 6.35623268747186338379024235562, 7.23459724988248786319351440132, 8.311887435735352063399447928054, 8.776340926333614439443780799572, 9.548127776243872162536191394401