| L(s) = 1 | + (−1.09 − 0.921i)2-s + (−1.39 + 1.02i)3-s + (0.00944 + 0.0535i)4-s + (1.35 + 0.492i)5-s + (2.47 + 0.169i)6-s + (0.173 − 0.984i)7-s + (−1.39 + 2.41i)8-s + (0.918 − 2.85i)9-s + (−1.03 − 1.78i)10-s + (1.39 − 0.507i)11-s + (−0.0678 − 0.0653i)12-s + (3.79 − 3.18i)13-s + (−1.09 + 0.921i)14-s + (−2.39 + 0.690i)15-s + (3.85 − 1.40i)16-s + (−3.76 − 6.52i)17-s + ⋯ |
| L(s) = 1 | + (−0.776 − 0.651i)2-s + (−0.808 + 0.588i)3-s + (0.00472 + 0.0267i)4-s + (0.604 + 0.220i)5-s + (1.01 + 0.0692i)6-s + (0.0656 − 0.372i)7-s + (−0.492 + 0.853i)8-s + (0.306 − 0.951i)9-s + (−0.326 − 0.565i)10-s + (0.420 − 0.152i)11-s + (−0.0195 − 0.0188i)12-s + (1.05 − 0.884i)13-s + (−0.293 + 0.246i)14-s + (−0.618 + 0.178i)15-s + (0.964 − 0.351i)16-s + (−0.913 − 1.58i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.524414 - 0.402995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524414 - 0.402995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.39 - 1.02i)T \) |
| 7 | \( 1 + (-0.173 + 0.984i)T \) |
| good | 2 | \( 1 + (1.09 + 0.921i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-1.35 - 0.492i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-1.39 + 0.507i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-3.79 + 3.18i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.76 + 6.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.13 + 7.15i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.31 - 7.45i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.675 - 0.567i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.22 - 6.93i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.822 - 1.42i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-7.19 + 6.03i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.81 - 0.662i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0880 - 0.499i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.788T + 53T^{2} \) |
| 59 | \( 1 + (6.05 + 2.20i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.792 - 4.49i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.02 - 6.73i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.83 + 3.18i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.51 - 6.09i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.16 - 3.49i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.26 + 1.89i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.57 + 2.73i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-14.0 + 5.11i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.73629182384851656515278987376, −11.21105279326462932786076306578, −10.46588286421079472712407849873, −9.495535421857004179638520958507, −8.951791530037467180933555323446, −7.14620181247992178988399904664, −5.89025091395452471964567573303, −4.92911536350102945315430164153, −3.07222732183618719553334789739, −0.948548656588192755079152846311,
1.63333156435089585676289866914, 4.15894438433559572629270608079, 6.12444574611504115438080800464, 6.26101473673194168345194906474, 7.73873014976004648524659284301, 8.602942122292592765704277888933, 9.598755467926141666695560269814, 10.74275832146660235127649377735, 11.88190164209021650510608085041, 12.71486047876081878196894141153
