L(s) = 1 | + (−0.766 − 0.642i)2-s + (−0.541 + 0.197i)3-s + (0.173 + 0.984i)4-s + (0.173 − 0.984i)5-s + (0.541 + 0.197i)6-s + (2.43 + 4.21i)7-s + (0.500 − 0.866i)8-s + (−2.04 + 1.71i)9-s + (−0.766 + 0.642i)10-s + (2.68 − 4.64i)11-s + (−0.288 − 0.499i)12-s + (3.62 + 1.31i)13-s + (0.844 − 4.79i)14-s + (0.100 + 0.567i)15-s + (−0.939 + 0.342i)16-s + (1.07 + 0.901i)17-s + ⋯ |
L(s) = 1 | + (−0.541 − 0.454i)2-s + (−0.312 + 0.113i)3-s + (0.0868 + 0.492i)4-s + (0.0776 − 0.440i)5-s + (0.221 + 0.0804i)6-s + (0.919 + 1.59i)7-s + (0.176 − 0.306i)8-s + (−0.681 + 0.571i)9-s + (−0.242 + 0.203i)10-s + (0.809 − 1.40i)11-s + (−0.0831 − 0.144i)12-s + (1.00 + 0.365i)13-s + (0.225 − 1.28i)14-s + (0.0258 + 0.146i)15-s + (−0.234 + 0.0855i)16-s + (0.260 + 0.218i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.922828 + 0.0310338i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.922828 + 0.0310338i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 5 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 + (-4.35 + 0.226i)T \) |
good | 3 | \( 1 + (0.541 - 0.197i)T + (2.29 - 1.92i)T^{2} \) |
| 7 | \( 1 + (-2.43 - 4.21i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.68 + 4.64i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.62 - 1.31i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (-1.07 - 0.901i)T + (2.95 + 16.7i)T^{2} \) |
| 23 | \( 1 + (-0.927 - 5.25i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (2.78 - 2.33i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (4.10 + 7.10i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 10.4T + 37T^{2} \) |
| 41 | \( 1 + (-1.79 + 0.652i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.256 + 1.45i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (2.39 - 2.00i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (0.312 + 1.77i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (5.61 + 4.70i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.40 - 7.99i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-6.46 + 5.42i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.04 + 5.93i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (-1.19 + 0.436i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-15.6 + 5.68i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.23 + 12.5i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.02 + 3.28i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (4.65 + 3.90i)T + (16.8 + 95.5i)T^{2} \) |
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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
−12.10756970357093362379447383656, −11.43348938348962959758222285795, −11.03548589706955339535689442075, −9.248084548007150535837542430317, −8.763359858151728520234251961420, −7.933223226435744206774218002023, −5.95484317344227868052988478228, −5.30569960067017941245904813418, −3.40924458962546139149572900868, −1.68757290958890336619381577458,
1.27946738305926572915466329657, 3.76794698874754528655155577042, 5.17243027402161210949226476946, 6.67306152057341175292245463683, 7.24347371535944515307559482891, 8.386373789764216950626320487157, 9.623546660637370482829511700993, 10.61522228913019980839473637761, 11.28531139384974613653763298578, 12.39737812534689824884952278055