| L(s) = 1 | + (0.266 − 0.733i)2-s + (1.16 + 1.28i)3-s + (1.06 + 0.893i)4-s + (−1.49 − 1.65i)5-s + (1.25 − 0.514i)6-s + (2.52 + 1.45i)7-s + (2.29 − 1.32i)8-s + (−0.277 + 2.98i)9-s + (−1.61 + 0.657i)10-s + (−3.52 + 2.03i)11-s + (0.0987 + 2.40i)12-s + (0.641 − 3.63i)13-s + (1.74 − 1.46i)14-s + (0.373 − 3.85i)15-s + (0.124 + 0.704i)16-s + (0.993 + 0.361i)17-s + ⋯ |
| L(s) = 1 | + (0.188 − 0.518i)2-s + (0.673 + 0.739i)3-s + (0.532 + 0.446i)4-s + (−0.670 − 0.741i)5-s + (0.510 − 0.209i)6-s + (0.955 + 0.551i)7-s + (0.810 − 0.467i)8-s + (−0.0924 + 0.995i)9-s + (−0.511 + 0.207i)10-s + (−1.06 + 0.612i)11-s + (0.0285 + 0.694i)12-s + (0.177 − 1.00i)13-s + (0.466 − 0.391i)14-s + (0.0964 − 0.995i)15-s + (0.0310 + 0.176i)16-s + (0.241 + 0.0877i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.87879 + 0.160595i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.87879 + 0.160595i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.16 - 1.28i)T \) |
| 5 | \( 1 + (1.49 + 1.65i)T \) |
| 19 | \( 1 + (1.64 + 4.03i)T \) |
| good | 2 | \( 1 + (-0.266 + 0.733i)T + (-1.53 - 1.28i)T^{2} \) |
| 7 | \( 1 + (-2.52 - 1.45i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.52 - 2.03i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.641 + 3.63i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.993 - 0.361i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (-1.88 - 1.58i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-3.47 + 1.26i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (7.63 + 4.40i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.63T + 37T^{2} \) |
| 41 | \( 1 + (0.771 + 4.37i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (6.90 + 8.22i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (10.8 - 3.95i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.566 - 0.675i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (6.30 + 2.29i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-5.24 - 4.39i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.61 - 1.31i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-1.20 + 1.01i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (4.16 - 0.734i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (10.4 - 1.84i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-7.48 + 12.9i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (2.14 - 12.1i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.3 - 3.75i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.71039360546810244334114378353, −11.02615736204400604150808705278, −10.16571505678448961450030166928, −8.873947421230978769622693873265, −8.017458821029360468455899763789, −7.51712118432564831644179786664, −5.29384247363218153432512700910, −4.54192033411095027283619324305, −3.29080736075765304700254651410, −2.11534239491704999360096336463,
1.68490939555719525428343171835, 3.14131365666286759744931741326, 4.65819780571177220039561560904, 6.15567036460599961153911192365, 7.01686242223003531923267607612, 7.81735560312950444967560390125, 8.382868365368078724171087622007, 10.09127008017686242848912935115, 11.07211380047224706235456848245, 11.57293751985235768664936381140
