| L(s) = 1 | + (−0.800 + 0.599i)2-s + (−0.0955 − 0.0356i)3-s + (0.281 − 0.959i)4-s + (−1.47 + 1.68i)5-s + (0.0978 − 0.0287i)6-s + (−2.15 + 0.468i)7-s + (0.349 + 0.936i)8-s + (−2.25 − 1.95i)9-s + (0.168 − 2.22i)10-s + (−1.11 + 0.160i)11-s + (−0.0611 + 0.0816i)12-s + (−0.355 + 1.63i)13-s + (1.44 − 1.66i)14-s + (0.200 − 0.108i)15-s + (−0.841 − 0.540i)16-s + (−1.55 + 2.83i)17-s + ⋯ |
| L(s) = 1 | + (−0.566 + 0.423i)2-s + (−0.0551 − 0.0205i)3-s + (0.140 − 0.479i)4-s + (−0.657 + 0.753i)5-s + (0.0399 − 0.0117i)6-s + (−0.814 + 0.177i)7-s + (0.123 + 0.331i)8-s + (−0.753 − 0.652i)9-s + (0.0533 − 0.705i)10-s + (−0.336 + 0.0483i)11-s + (−0.0176 + 0.0235i)12-s + (−0.0985 + 0.453i)13-s + (0.386 − 0.445i)14-s + (0.0518 − 0.0280i)15-s + (−0.210 − 0.135i)16-s + (−0.376 + 0.688i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 230 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00716880 - 0.218128i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00716880 - 0.218128i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.800 - 0.599i)T \) |
| 5 | \( 1 + (1.47 - 1.68i)T \) |
| 23 | \( 1 + (3.52 - 3.24i)T \) |
| good | 3 | \( 1 + (0.0955 + 0.0356i)T + (2.26 + 1.96i)T^{2} \) |
| 7 | \( 1 + (2.15 - 0.468i)T + (6.36 - 2.90i)T^{2} \) |
| 11 | \( 1 + (1.11 - 0.160i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (0.355 - 1.63i)T + (-11.8 - 5.40i)T^{2} \) |
| 17 | \( 1 + (1.55 - 2.83i)T + (-9.19 - 14.3i)T^{2} \) |
| 19 | \( 1 + (4.49 + 1.32i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-0.667 - 2.27i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.69 + 5.90i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.118 - 1.65i)T + (-36.6 + 5.26i)T^{2} \) |
| 41 | \( 1 + (-4.53 - 5.23i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.209 + 0.562i)T + (-32.4 - 28.1i)T^{2} \) |
| 47 | \( 1 + (-2.27 - 2.27i)T + 47iT^{2} \) |
| 53 | \( 1 + (-2.25 - 10.3i)T + (-48.2 + 22.0i)T^{2} \) |
| 59 | \( 1 + (-2.64 - 4.11i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-9.76 - 4.45i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (4.04 + 5.39i)T + (-18.8 + 64.2i)T^{2} \) |
| 71 | \( 1 + (-2.33 + 16.2i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-4.27 + 2.33i)T + (39.4 - 61.4i)T^{2} \) |
| 79 | \( 1 + (10.1 - 6.50i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (0.799 - 0.0572i)T + (82.1 - 11.8i)T^{2} \) |
| 89 | \( 1 + (4.52 + 9.90i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (11.3 + 0.809i)T + (96.0 + 13.8i)T^{2} \) |
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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
−12.54316004768760466317836800446, −11.57430203875486676912732953867, −10.73820462456243425535805985044, −9.714890298760981045499282084722, −8.742342166352742775079104642867, −7.75564072034333323652071261604, −6.59083044154876566918453331888, −6.00273043751758077238220459883, −4.09738174132442817801895044017, −2.68742925305132467519388734094,
0.20130751450998712522374884626, 2.56623990796033619443834989063, 4.01035435252080290535388415745, 5.35478992678059833581751492567, 6.82303620931399201915902280598, 8.103124764062527294369761291345, 8.630605496046728939564853674884, 9.843495728638606603078213453281, 10.72794164780391054998790789776, 11.66567503801720423259877046243
