| L(s) = 1 | + (2.98 − 0.800i)2-s + (1.07 − 5.08i)3-s + (1.36 − 0.787i)4-s + (6.51 + 6.51i)5-s + (−0.854 − 16.0i)6-s + (4.63 − 17.2i)7-s + (−14.0 + 14.0i)8-s + (−24.6 − 10.9i)9-s + (24.6 + 14.2i)10-s + (10.6 + 39.7i)11-s + (−2.53 − 7.77i)12-s + (−1.70 + 46.8i)13-s − 55.3i·14-s + (40.1 − 26.1i)15-s + (−37.0 + 64.1i)16-s + (−18.9 − 32.7i)17-s + ⋯ |
| L(s) = 1 | + (1.05 − 0.283i)2-s + (0.207 − 0.978i)3-s + (0.170 − 0.0984i)4-s + (0.582 + 0.582i)5-s + (−0.0581 − 1.09i)6-s + (0.250 − 0.933i)7-s + (−0.621 + 0.621i)8-s + (−0.914 − 0.405i)9-s + (0.780 + 0.450i)10-s + (0.292 + 1.09i)11-s + (−0.0609 − 0.187i)12-s + (−0.0363 + 0.999i)13-s − 1.05i·14-s + (0.690 − 0.449i)15-s + (−0.579 + 1.00i)16-s + (−0.270 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 39 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.95969 - 0.836897i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.95969 - 0.836897i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.07 + 5.08i)T \) |
| 13 | \( 1 + (1.70 - 46.8i)T \) |
| good | 2 | \( 1 + (-2.98 + 0.800i)T + (6.92 - 4i)T^{2} \) |
| 5 | \( 1 + (-6.51 - 6.51i)T + 125iT^{2} \) |
| 7 | \( 1 + (-4.63 + 17.2i)T + (-297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-10.6 - 39.7i)T + (-1.15e3 + 665.5i)T^{2} \) |
| 17 | \( 1 + (18.9 + 32.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-11.2 - 3.02i)T + (5.94e3 + 3.42e3i)T^{2} \) |
| 23 | \( 1 + (-36.7 + 63.6i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + (261. + 150. i)T + (1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 + (-144. + 144. i)T - 2.97e4iT^{2} \) |
| 37 | \( 1 + (53.9 - 14.4i)T + (4.38e4 - 2.53e4i)T^{2} \) |
| 41 | \( 1 + (-297. + 79.6i)T + (5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (-136. + 78.8i)T + (3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + (260. - 260. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 - 411. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (442. + 118. i)T + (1.77e5 + 1.02e5i)T^{2} \) |
| 61 | \( 1 + (-36.3 - 62.9i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-231. - 862. i)T + (-2.60e5 + 1.50e5i)T^{2} \) |
| 71 | \( 1 + (-180. + 672. i)T + (-3.09e5 - 1.78e5i)T^{2} \) |
| 73 | \( 1 + (366. + 366. i)T + 3.89e5iT^{2} \) |
| 79 | \( 1 - 764.T + 4.93e5T^{2} \) |
| 83 | \( 1 + (-332. - 332. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + (242. + 904. i)T + (-6.10e5 + 3.52e5i)T^{2} \) |
| 97 | \( 1 + (912. + 244. i)T + (7.90e5 + 4.56e5i)T^{2} \) |
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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
−14.87184128685238750655257501575, −14.07785871618074334750147853900, −13.43644213636894799568075557009, −12.28200758993419500515670079590, −11.17751487679867734156637967458, −9.363538932418017193260198219393, −7.47409686723304734777266211483, −6.26353233879456350677148574550, −4.30285987214667881243824494839, −2.29563421195787856140182924645,
3.36383995662732898520129441650, 5.15904063693218065335559400039, 5.80625989120207329345458785219, 8.600995269605257792999213310237, 9.504758264720182558955423242639, 11.15300271438292429994393619852, 12.66712700446289954879530806412, 13.69288227687485157804339958561, 14.78291555960063500272308304694, 15.54180214070191820240436903032
