| L(s) = 1 | + (1.28 − 3.54i)2-s + (−7.81 − 6.55i)4-s + (6.09 − 5.11i)5-s + (−2.17 + 3.76i)7-s + (−20.2 + 11.6i)8-s + (−10.2 − 28.1i)10-s + (−4.81 − 8.34i)11-s + (11.2 + 1.98i)13-s + (10.5 + 12.5i)14-s + (8.21 + 46.5i)16-s + (−3.02 − 1.09i)17-s + (4.60 + 18.4i)19-s − 81.1·20-s + (−35.7 + 6.30i)22-s + (19.2 + 16.1i)23-s + ⋯ |
| L(s) = 1 | + (0.644 − 1.77i)2-s + (−1.95 − 1.63i)4-s + (1.21 − 1.02i)5-s + (−0.310 + 0.538i)7-s + (−2.53 + 1.46i)8-s + (−1.02 − 2.81i)10-s + (−0.437 − 0.758i)11-s + (0.867 + 0.152i)13-s + (0.753 + 0.897i)14-s + (0.513 + 2.91i)16-s + (−0.177 − 0.0646i)17-s + (0.242 + 0.970i)19-s − 4.05·20-s + (−1.62 + 0.286i)22-s + (0.835 + 0.701i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.998 - 0.0499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0502964 + 2.01252i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0502964 + 2.01252i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (-4.60 - 18.4i)T \) |
| good | 2 | \( 1 + (-1.28 + 3.54i)T + (-3.06 - 2.57i)T^{2} \) |
| 5 | \( 1 + (-6.09 + 5.11i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (2.17 - 3.76i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (4.81 + 8.34i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-11.2 - 1.98i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (3.02 + 1.09i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (-19.2 - 16.1i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (7.54 + 20.7i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-3.93 - 2.27i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 38.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (20.0 - 3.53i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (-49.0 + 41.1i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-5.05 + 1.83i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-52.6 + 62.6i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (23.5 - 64.7i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-11.0 - 9.24i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-6.23 - 17.1i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-8.50 - 10.1i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-7.50 - 42.5i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (49.5 - 8.72i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (75.4 - 130. i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (56.0 + 9.88i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (45.3 - 124. i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−12.15061727545253703200854354931, −11.13710052965946964756011391991, −10.15899554127259949284799569068, −9.276216673615671136631516534343, −8.639003480276170740239617980450, −5.81926443481056231470712255967, −5.39972080935587532191999053603, −3.86663935946896882763193041184, −2.42990093753306713811245896980, −1.13964168092659607276247062014,
3.05852152122589224974144865070, 4.64846777765136479839638492299, 5.82967686119397511855390927107, 6.71299213779360993841868080691, 7.29624805814975725260322101242, 8.724177498917951047718096154311, 9.786294521886166982144474360081, 10.86929011836795259887419793168, 12.76177388844400547834024036672, 13.42637569625583570228310312807
