| L(s) = 1 | + (0.710 − 1.55i)2-s + (−0.841 + 0.540i)3-s + (−0.605 − 0.698i)4-s + (−0.0752 + 0.523i)5-s + (0.243 + 1.69i)6-s + (1.63 − 3.57i)7-s + (1.76 − 0.518i)8-s + (0.415 − 0.909i)9-s + (0.760 + 0.488i)10-s + (0.411 − 2.86i)11-s + (0.886 + 0.260i)12-s + (−0.394 − 0.115i)13-s + (−4.40 − 5.07i)14-s + (−0.219 − 0.481i)15-s + (0.710 − 4.94i)16-s + (−3.38 + 3.90i)17-s + ⋯ |
| L(s) = 1 | + (0.502 − 1.09i)2-s + (−0.485 + 0.312i)3-s + (−0.302 − 0.349i)4-s + (−0.0336 + 0.234i)5-s + (0.0993 + 0.690i)6-s + (0.617 − 1.35i)7-s + (0.624 − 0.183i)8-s + (0.138 − 0.303i)9-s + (0.240 + 0.154i)10-s + (0.124 − 0.863i)11-s + (0.255 + 0.0751i)12-s + (−0.109 − 0.0321i)13-s + (−1.17 − 1.35i)14-s + (−0.0567 − 0.124i)15-s + (0.177 − 1.23i)16-s + (−0.820 + 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10053 - 0.961681i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10053 - 0.961681i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.841 - 0.540i)T \) |
| 67 | \( 1 + (-3.73 - 7.28i)T \) |
| good | 2 | \( 1 + (-0.710 + 1.55i)T + (-1.30 - 1.51i)T^{2} \) |
| 5 | \( 1 + (0.0752 - 0.523i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.63 + 3.57i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.411 + 2.86i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (0.394 + 0.115i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (3.38 - 3.90i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-3.54 - 7.76i)T + (-12.4 + 14.3i)T^{2} \) |
| 23 | \( 1 + (3.13 - 2.01i)T + (9.55 - 20.9i)T^{2} \) |
| 29 | \( 1 + 8.48T + 29T^{2} \) |
| 31 | \( 1 + (-4.67 + 1.37i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + 7.91T + 37T^{2} \) |
| 41 | \( 1 + (-3.45 + 3.98i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (0.524 - 0.605i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (6.60 - 4.24i)T + (19.5 - 42.7i)T^{2} \) |
| 53 | \( 1 + (-1.09 - 1.26i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.04 + 0.599i)T + (49.6 - 31.8i)T^{2} \) |
| 61 | \( 1 + (-1.18 - 8.26i)T + (-58.5 + 17.1i)T^{2} \) |
| 71 | \( 1 + (-10.2 - 11.8i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (1.35 + 9.42i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (6.30 + 1.85i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (0.0668 - 0.465i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (-0.897 - 0.576i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + 4.69T + 97T^{2} \) |
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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
−12.02373857668890693325732543926, −11.15060808288511354980482431615, −10.66219282042210127068099762193, −9.864174443195111546025671397629, −8.149745219958127196550958726836, −7.10248887973755400803451287476, −5.61764894146911176829713890898, −4.19447852184052970971515235805, −3.54165385602132846653573765446, −1.47340306884366396714800981690,
2.19406035437943305201216889281, 4.83691584012835097990752196572, 5.15519736242656880744574943953, 6.50398761735825883957746128302, 7.25650832239852911953304342065, 8.443820874992036560903009310393, 9.459776794359750434201118859249, 11.05274740655152620115102306928, 11.80296677058279837918044977081, 12.77704611263186767485538610826
