| 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 | \( 1 + 4.69T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.77704611263186767485538610826, −11.80296677058279837918044977081, −11.05274740655152620115102306928, −9.459776794359750434201118859249, −8.443820874992036560903009310393, −7.25650832239852911953304342065, −6.50398761735825883957746128302, −5.15519736242656880744574943953, −4.83691584012835097990752196572, −2.19406035437943305201216889281,
1.47340306884366396714800981690, 3.54165385602132846653573765446, 4.19447852184052970971515235805, 5.61764894146911176829713890898, 7.10248887973755400803451287476, 8.149745219958127196550958726836, 9.864174443195111546025671397629, 10.66219282042210127068099762193, 11.15060808288511354980482431615, 12.02373857668890693325732543926
