| L(s) = 1 | + (−0.669 − 1.24i)2-s + (−0.634 − 0.773i)3-s + (−1.10 + 1.66i)4-s + (−2.98 − 0.904i)5-s + (−0.537 + 1.30i)6-s + (1.28 − 0.254i)7-s + (2.81 + 0.256i)8-s + (−0.195 + 0.980i)9-s + (0.870 + 4.32i)10-s + (−4.61 + 0.454i)11-s + (1.98 − 0.206i)12-s + (0.573 + 1.89i)13-s + (−1.17 − 1.42i)14-s + (1.19 + 2.87i)15-s + (−1.56 − 3.67i)16-s + (−1.01 + 2.44i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 − 0.880i)2-s + (−0.366 − 0.446i)3-s + (−0.551 + 0.834i)4-s + (−1.33 − 0.404i)5-s + (−0.219 + 0.533i)6-s + (0.484 − 0.0963i)7-s + (0.995 + 0.0905i)8-s + (−0.0650 + 0.326i)9-s + (0.275 + 1.36i)10-s + (−1.39 + 0.137i)11-s + (0.574 − 0.0594i)12-s + (0.159 + 0.524i)13-s + (−0.314 − 0.380i)14-s + (0.307 + 0.743i)15-s + (−0.391 − 0.919i)16-s + (−0.245 + 0.593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.319895 + 0.126631i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.319895 + 0.126631i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 + 1.24i)T \) |
| 3 | \( 1 + (0.634 + 0.773i)T \) |
| good | 5 | \( 1 + (2.98 + 0.904i)T + (4.15 + 2.77i)T^{2} \) |
| 7 | \( 1 + (-1.28 + 0.254i)T + (6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (4.61 - 0.454i)T + (10.7 - 2.14i)T^{2} \) |
| 13 | \( 1 + (-0.573 - 1.89i)T + (-10.8 + 7.22i)T^{2} \) |
| 17 | \( 1 + (1.01 - 2.44i)T + (-12.0 - 12.0i)T^{2} \) |
| 19 | \( 1 + (-4.25 - 2.27i)T + (10.5 + 15.7i)T^{2} \) |
| 23 | \( 1 + (-4.72 - 7.07i)T + (-8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (-0.221 + 2.25i)T + (-28.4 - 5.65i)T^{2} \) |
| 31 | \( 1 + (0.564 - 0.564i)T - 31iT^{2} \) |
| 37 | \( 1 + (0.767 + 1.43i)T + (-20.5 + 30.7i)T^{2} \) |
| 41 | \( 1 + (1.28 - 0.855i)T + (15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (5.67 - 6.91i)T + (-8.38 - 42.1i)T^{2} \) |
| 47 | \( 1 + (0.668 + 0.277i)T + (33.2 + 33.2i)T^{2} \) |
| 53 | \( 1 + (0.988 + 10.0i)T + (-51.9 + 10.3i)T^{2} \) |
| 59 | \( 1 + (3.27 - 10.8i)T + (-49.0 - 32.7i)T^{2} \) |
| 61 | \( 1 + (4.64 - 3.81i)T + (11.9 - 59.8i)T^{2} \) |
| 67 | \( 1 + (5.87 - 4.82i)T + (13.0 - 65.7i)T^{2} \) |
| 71 | \( 1 + (-2.72 - 13.7i)T + (-65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (11.0 + 2.19i)T + (67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (-8.36 + 3.46i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 + (2.67 - 5.00i)T + (-46.1 - 69.0i)T^{2} \) |
| 89 | \( 1 + (-6.59 + 9.86i)T + (-34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (9.66 - 9.66i)T - 97iT^{2} \) |
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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
−11.48971602082183073102425394474, −10.84069594739383586464423938770, −9.752441285875492281288075812107, −8.516127827191379502521952340538, −7.84643975187967681696412328430, −7.26199227943510665650854143459, −5.32811881801361272469884741781, −4.36761425348770918176760607568, −3.16018159122721671166755342057, −1.44126477777630625951977613792,
0.30157084076836254464317719191, 3.13029350471851939776577181167, 4.69178034398365402319855950292, 5.24444208419685467622863768803, 6.68879051530795703870967417523, 7.59873264380009525803188337321, 8.219900994226153285843757776434, 9.215602349551588075377087901235, 10.56540916931235107202302065726, 10.86026487151990578913106825769
