| L(s) = 1 | + (−0.262 − 0.361i)2-s + (2.64 + 0.860i)3-s + (0.556 − 1.71i)4-s + (0.0123 + 2.23i)5-s + (−0.384 − 1.18i)6-s − 4.32i·7-s + (−1.61 + 0.525i)8-s + (3.84 + 2.79i)9-s + (0.805 − 0.592i)10-s + (−0.742 + 0.539i)11-s + (2.94 − 4.05i)12-s + (1.76 − 2.43i)13-s + (−1.56 + 1.13i)14-s + (−1.89 + 5.93i)15-s + (−2.29 − 1.66i)16-s + (2.92 − 0.948i)17-s + ⋯ |
| L(s) = 1 | + (−0.185 − 0.255i)2-s + (1.52 + 0.496i)3-s + (0.278 − 0.856i)4-s + (0.00553 + 0.999i)5-s + (−0.157 − 0.483i)6-s − 1.63i·7-s + (−0.571 + 0.185i)8-s + (1.28 + 0.931i)9-s + (0.254 − 0.187i)10-s + (−0.223 + 0.162i)11-s + (0.850 − 1.17i)12-s + (0.490 − 0.675i)13-s + (−0.418 + 0.303i)14-s + (−0.488 + 1.53i)15-s + (−0.574 − 0.417i)16-s + (0.708 − 0.230i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.12535 - 0.533182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.12535 - 0.533182i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (-0.0123 - 2.23i)T \) |
| 19 | \( 1 + (-0.309 - 0.951i)T \) |
| good | 2 | \( 1 + (0.262 + 0.361i)T + (-0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (-2.64 - 0.860i)T + (2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 4.32iT - 7T^{2} \) |
| 11 | \( 1 + (0.742 - 0.539i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-1.76 + 2.43i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.92 + 0.948i)T + (13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (-3.67 - 5.06i)T + (-7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (1.90 - 5.85i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.458 + 1.40i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (3.89 - 5.36i)T + (-11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-1.95 - 1.42i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 9.63iT - 43T^{2} \) |
| 47 | \( 1 + (10.0 + 3.27i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (4.90 + 1.59i)T + (42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (5.68 + 4.12i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-6.29 + 4.57i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (-8.04 + 2.61i)T + (54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (-3.62 + 11.1i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-7.55 - 10.3i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-3.35 + 10.3i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (10.0 - 3.27i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (3.85 - 2.80i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (0.493 + 0.160i)T + (78.4 + 57.0i)T^{2} \) |
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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
−10.75466263290498881349083279711, −9.861483100495049013450948757998, −9.641987302429453763307437073990, −8.153780856084427283859849479420, −7.45315136900965549633857022063, −6.57323012373801888846337083478, −5.04880365482599729906602409811, −3.57261171709917307932723989389, −3.07912058217488582457784872335, −1.51335967138465575178449618317,
1.96026313754116632374652129347, 2.84164846071382192802203691505, 3.97958120971127140482238182459, 5.51019892078526009032998233569, 6.70537675432843367603608888158, 7.85237282368437287115593009776, 8.484793319708933669503937725332, 8.891403949638187998874042688987, 9.530706577320627626568212889882, 11.37657343596796962449946349164
