| L(s) = 1 | + (0.326 + 1.37i)2-s + (2.81 − 1.02i)3-s + (−1.78 + 0.897i)4-s + (−1.67 + 1.24i)5-s + (2.33 + 3.54i)6-s + (11.2 − 7.37i)7-s + (−1.81 − 2.16i)8-s + (6.89 − 5.79i)9-s + (−2.26 − 1.90i)10-s + (−0.585 + 5.01i)11-s + (−4.11 + 4.36i)12-s + (6.22 + 20.7i)13-s + (13.8 + 13.0i)14-s + (−3.44 + 5.24i)15-s + (2.38 − 3.20i)16-s + (−14.1 − 2.49i)17-s + ⋯ |
| L(s) = 1 | + (0.163 + 0.688i)2-s + (0.939 − 0.342i)3-s + (−0.446 + 0.224i)4-s + (−0.335 + 0.249i)5-s + (0.388 + 0.590i)6-s + (1.60 − 1.05i)7-s + (−0.227 − 0.270i)8-s + (0.765 − 0.643i)9-s + (−0.226 − 0.190i)10-s + (−0.0532 + 0.455i)11-s + (−0.342 + 0.363i)12-s + (0.478 + 1.59i)13-s + (0.986 + 0.930i)14-s + (−0.229 + 0.349i)15-s + (0.149 − 0.200i)16-s + (−0.833 − 0.146i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.855 - 0.517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.10578 + 0.587869i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.10578 + 0.587869i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.326 - 1.37i)T \) |
| 3 | \( 1 + (-2.81 + 1.02i)T \) |
| good | 5 | \( 1 + (1.67 - 1.24i)T + (7.17 - 23.9i)T^{2} \) |
| 7 | \( 1 + (-11.2 + 7.37i)T + (19.4 - 44.9i)T^{2} \) |
| 11 | \( 1 + (0.585 - 5.01i)T + (-117. - 27.9i)T^{2} \) |
| 13 | \( 1 + (-6.22 - 20.7i)T + (-141. + 92.8i)T^{2} \) |
| 17 | \( 1 + (14.1 + 2.49i)T + (271. + 98.8i)T^{2} \) |
| 19 | \( 1 + (1.82 + 10.3i)T + (-339. + 123. i)T^{2} \) |
| 23 | \( 1 + (-11.9 + 18.1i)T + (-209. - 485. i)T^{2} \) |
| 29 | \( 1 + (34.6 - 32.6i)T + (48.8 - 839. i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 29.0i)T + (-954. - 111. i)T^{2} \) |
| 37 | \( 1 + (9.16 - 3.33i)T + (1.04e3 - 879. i)T^{2} \) |
| 41 | \( 1 + (-0.224 + 0.948i)T + (-1.50e3 - 754. i)T^{2} \) |
| 43 | \( 1 + (21.0 + 48.8i)T + (-1.26e3 + 1.34e3i)T^{2} \) |
| 47 | \( 1 + (50.4 - 2.93i)T + (2.19e3 - 256. i)T^{2} \) |
| 53 | \( 1 + (37.4 + 21.6i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-12.6 - 108. i)T + (-3.38e3 + 802. i)T^{2} \) |
| 61 | \( 1 + (16.0 + 8.06i)T + (2.22e3 + 2.98e3i)T^{2} \) |
| 67 | \( 1 + (51.6 - 54.7i)T + (-261. - 4.48e3i)T^{2} \) |
| 71 | \( 1 + (-61.4 + 73.2i)T + (-875. - 4.96e3i)T^{2} \) |
| 73 | \( 1 + (24.8 - 20.8i)T + (925. - 5.24e3i)T^{2} \) |
| 79 | \( 1 + (21.4 - 5.09i)T + (5.57e3 - 2.80e3i)T^{2} \) |
| 83 | \( 1 + (0.216 + 0.912i)T + (-6.15e3 + 3.09e3i)T^{2} \) |
| 89 | \( 1 + (-21.9 - 26.1i)T + (-1.37e3 + 7.80e3i)T^{2} \) |
| 97 | \( 1 + (25.5 - 34.3i)T + (-2.69e3 - 9.01e3i)T^{2} \) |
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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
−13.21006944940193272567797114037, −11.68944457564747452265730385654, −10.83613671553437871579503046449, −9.244406301313989396719651534778, −8.428664497028315068118002614843, −7.32567655976015253648721317825, −6.88590576968562703547210200492, −4.72014079302485519967494017377, −3.92472957286215621493219985312, −1.76474125419946730782216461812,
1.78267070428590658726549007560, 3.18210512334497939240356378092, 4.58210760190813016257818171038, 5.60392700078245085156819915923, 8.109735806467491122738997344710, 8.233833380062566181052809350148, 9.419433158120115094609874523896, 10.73456048998597345494106081752, 11.41828886579404114494720671491, 12.58332947907388874040393656191
