| L(s) = 1 | + (−0.766 + 1.18i)2-s + (−1.39 + 0.137i)3-s + (−0.823 − 1.82i)4-s + (1.67 − 3.12i)5-s + (0.906 − 1.76i)6-s + (0.194 − 0.979i)7-s + (2.79 + 0.419i)8-s + (−1.01 + 0.201i)9-s + (2.43 + 4.38i)10-s + (0.362 − 0.297i)11-s + (1.39 + 2.42i)12-s + (5.08 − 2.71i)13-s + (1.01 + 0.982i)14-s + (−1.90 + 4.59i)15-s + (−2.64 + 3.00i)16-s + (−0.701 − 1.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.542 + 0.840i)2-s + (−0.805 + 0.0793i)3-s + (−0.411 − 0.911i)4-s + (0.747 − 1.39i)5-s + (0.370 − 0.719i)6-s + (0.0736 − 0.370i)7-s + (0.988 + 0.148i)8-s + (−0.338 + 0.0672i)9-s + (0.769 + 1.38i)10-s + (0.109 − 0.0896i)11-s + (0.403 + 0.701i)12-s + (1.40 − 0.753i)13-s + (0.271 + 0.262i)14-s + (−0.491 + 1.18i)15-s + (−0.660 + 0.750i)16-s + (−0.170 − 0.410i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.660579 - 0.143966i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.660579 - 0.143966i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 - 1.18i)T \) |
| good | 3 | \( 1 + (1.39 - 0.137i)T + (2.94 - 0.585i)T^{2} \) |
| 5 | \( 1 + (-1.67 + 3.12i)T + (-2.77 - 4.15i)T^{2} \) |
| 7 | \( 1 + (-0.194 + 0.979i)T + (-6.46 - 2.67i)T^{2} \) |
| 11 | \( 1 + (-0.362 + 0.297i)T + (2.14 - 10.7i)T^{2} \) |
| 13 | \( 1 + (-5.08 + 2.71i)T + (7.22 - 10.8i)T^{2} \) |
| 17 | \( 1 + (0.701 + 1.69i)T + (-12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (-0.364 - 0.110i)T + (15.7 + 10.5i)T^{2} \) |
| 23 | \( 1 + (6.09 + 4.07i)T + (8.80 + 21.2i)T^{2} \) |
| 29 | \( 1 + (2.16 - 2.64i)T + (-5.65 - 28.4i)T^{2} \) |
| 31 | \( 1 + (-6.95 - 6.95i)T + 31iT^{2} \) |
| 37 | \( 1 + (0.132 + 0.436i)T + (-30.7 + 20.5i)T^{2} \) |
| 41 | \( 1 + (3.08 - 4.62i)T + (-15.6 - 37.8i)T^{2} \) |
| 43 | \( 1 + (-2.80 - 0.276i)T + (42.1 + 8.38i)T^{2} \) |
| 47 | \( 1 + (8.00 - 3.31i)T + (33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-8.24 - 10.0i)T + (-10.3 + 51.9i)T^{2} \) |
| 59 | \( 1 + (5.72 + 3.05i)T + (32.7 + 49.0i)T^{2} \) |
| 61 | \( 1 + (-0.179 - 1.81i)T + (-59.8 + 11.9i)T^{2} \) |
| 67 | \( 1 + (0.243 + 2.47i)T + (-65.7 + 13.0i)T^{2} \) |
| 71 | \( 1 + (-14.3 - 2.85i)T + (65.5 + 27.1i)T^{2} \) |
| 73 | \( 1 + (-1.12 - 5.63i)T + (-67.4 + 27.9i)T^{2} \) |
| 79 | \( 1 + (1.06 + 0.443i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-4.47 + 14.7i)T + (-69.0 - 46.1i)T^{2} \) |
| 89 | \( 1 + (11.2 - 7.52i)T + (34.0 - 82.2i)T^{2} \) |
| 97 | \( 1 + (-1.53 - 1.53i)T + 97iT^{2} \) |
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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
−13.48213717073942826380403602577, −12.33808725449340143361851193676, −10.96047560203678615852904285168, −10.06787992387036214617396140505, −8.830352040574722745228298775385, −8.182844948153348973255150808244, −6.37215820074939902874984749232, −5.62151296893654361400406332720, −4.63211809988441646876116492907, −1.03130998518253711823781677430,
2.12981158143748007574484360709, 3.71449897480664657987646458330, 5.83267561825445283796565631127, 6.70093391890964898094608878120, 8.278925615580897697629443247332, 9.560466092037535343089812522257, 10.49924738695237492359419916064, 11.35713002524675823483361444620, 11.85122053247400249788329795231, 13.41375608575706204420517441284
