| L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (−1.76 + 0.517i)5-s + (−0.415 − 0.909i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (1.20 − 1.38i)10-s + (2.20 + 1.41i)11-s + (0.841 + 0.540i)12-s + (2.03 − 2.34i)13-s + (−0.959 − 0.281i)14-s + (−0.261 − 1.81i)15-s + (−0.654 − 0.755i)16-s + (2.32 + 5.08i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (−0.787 + 0.231i)5-s + (−0.169 − 0.371i)6-s + (0.247 + 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (0.380 − 0.438i)10-s + (0.665 + 0.427i)11-s + (0.242 + 0.156i)12-s + (0.564 − 0.651i)13-s + (−0.256 − 0.0752i)14-s + (−0.0674 − 0.469i)15-s + (−0.163 − 0.188i)16-s + (0.563 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.251854 + 0.792200i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.251854 + 0.792200i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (-1.97 - 4.37i)T \) |
| good | 5 | \( 1 + (1.76 - 0.517i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (-2.20 - 1.41i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-2.03 + 2.34i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.32 - 5.08i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (-0.745 + 1.63i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (0.679 + 1.48i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.147 - 1.02i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (7.72 + 2.26i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (4.24 - 1.24i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (1.13 - 7.88i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 + (-6.55 - 7.56i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (5.67 - 6.55i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.266 + 1.85i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (-11.1 + 7.18i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (6.12 - 3.93i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (4.58 - 10.0i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-6.17 + 7.12i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (8.40 + 2.46i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (0.235 - 1.63i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (7.77 - 2.28i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.28834932767227925008419851209, −9.479616967340352716621217601421, −8.631421274440945060314526761822, −7.958931009694570389164190087540, −7.13747751352441890954405392672, −6.08993075043027933528659294033, −5.26622973123381109539822304310, −4.07997221829474842558989753267, −3.22502384551208088994709478170, −1.46247607467987234076839862804,
0.51834107816506296452055988230, 1.71555309270209008350354490075, 3.20917143146738791679611876244, 4.10237064667254133785447650903, 5.30403271590055308648937153544, 6.64866259936154181904122346785, 7.18579041635816275565014966843, 8.209095499980009862280959656342, 8.667191033631250630571796237461, 9.620353620617562160762260673823
