| 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
−11.66567503801720423259877046243, −10.72794164780391054998790789776, −9.843495728638606603078213453281, −8.630605496046728939564853674884, −8.103124764062527294369761291345, −6.82303620931399201915902280598, −5.35478992678059833581751492567, −4.01035435252080290535388415745, −2.56623990796033619443834989063, −0.20130751450998712522374884626,
2.68742925305132467519388734094, 4.09738174132442817801895044017, 6.00273043751758077238220459883, 6.59083044154876566918453331888, 7.75564072034333323652071261604, 8.742342166352742775079104642867, 9.714890298760981045499282084722, 10.73820462456243425535805985044, 11.57430203875486676912732953867, 12.54316004768760466317836800446
