| L(s) = 1 | + (2.79 − 0.442i)2-s + (14.1 + 5.85i)3-s + (7.60 − 2.47i)4-s + (−12.1 + 23.8i)5-s + (42.0 + 10.0i)6-s + (−7.81 + 1.87i)7-s + (20.1 − 10.2i)8-s + (108. + 108. i)9-s + (−23.4 + 72.1i)10-s + (−2.97 − 2.54i)11-s + (121. + 9.59i)12-s + (−50.2 − 82.0i)13-s + (−21.0 + 8.70i)14-s + (−311. + 266. i)15-s + (51.7 − 37.6i)16-s + (−26.6 − 339. i)17-s + ⋯ |
| L(s) = 1 | + (0.698 − 0.110i)2-s + (1.56 + 0.650i)3-s + (0.475 − 0.154i)4-s + (−0.486 + 0.955i)5-s + (1.16 + 0.280i)6-s + (−0.159 + 0.0382i)7-s + (0.315 − 0.160i)8-s + (1.33 + 1.33i)9-s + (−0.234 + 0.721i)10-s + (−0.0246 − 0.0210i)11-s + (0.846 + 0.0666i)12-s + (−0.297 − 0.485i)13-s + (−0.107 + 0.0443i)14-s + (−1.38 + 1.18i)15-s + (0.202 − 0.146i)16-s + (−0.0923 − 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.709 - 0.704i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(3.38127 + 1.39347i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.38127 + 1.39347i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.79 + 0.442i)T \) |
| 41 | \( 1 + (589. - 1.57e3i)T \) |
| good | 3 | \( 1 + (-14.1 - 5.85i)T + (57.2 + 57.2i)T^{2} \) |
| 5 | \( 1 + (12.1 - 23.8i)T + (-367. - 505. i)T^{2} \) |
| 7 | \( 1 + (7.81 - 1.87i)T + (2.13e3 - 1.09e3i)T^{2} \) |
| 11 | \( 1 + (2.97 + 2.54i)T + (2.29e3 + 1.44e4i)T^{2} \) |
| 13 | \( 1 + (50.2 + 82.0i)T + (-1.29e4 + 2.54e4i)T^{2} \) |
| 17 | \( 1 + (26.6 + 339. i)T + (-8.24e4 + 1.30e4i)T^{2} \) |
| 19 | \( 1 + (-122. + 199. i)T + (-5.91e4 - 1.16e5i)T^{2} \) |
| 23 | \( 1 + (-53.4 + 73.5i)T + (-8.64e4 - 2.66e5i)T^{2} \) |
| 29 | \( 1 + (5.10 - 64.8i)T + (-6.98e5 - 1.10e5i)T^{2} \) |
| 31 | \( 1 + (-1.44e3 - 470. i)T + (7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (815. + 2.51e3i)T + (-1.51e6 + 1.10e6i)T^{2} \) |
| 43 | \( 1 + (-1.12e3 + 178. i)T + (3.25e6 - 1.05e6i)T^{2} \) |
| 47 | \( 1 + (4.06e3 + 976. i)T + (4.34e6 + 2.21e6i)T^{2} \) |
| 53 | \( 1 + (3.94e3 + 310. i)T + (7.79e6 + 1.23e6i)T^{2} \) |
| 59 | \( 1 + (-3.92e3 - 2.85e3i)T + (3.74e6 + 1.15e7i)T^{2} \) |
| 61 | \( 1 + (32.3 - 204. i)T + (-1.31e7 - 4.27e6i)T^{2} \) |
| 67 | \( 1 + (2.60e3 + 3.04e3i)T + (-3.15e6 + 1.99e7i)T^{2} \) |
| 71 | \( 1 + (1.29e3 - 1.51e3i)T + (-3.97e6 - 2.50e7i)T^{2} \) |
| 73 | \( 1 + (-379. + 379. i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + (3.73e3 - 9.02e3i)T + (-2.75e7 - 2.75e7i)T^{2} \) |
| 83 | \( 1 - 3.20e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-5.79e3 + 1.39e3i)T + (5.59e7 - 2.84e7i)T^{2} \) |
| 97 | \( 1 + (-4.45e3 + 3.80e3i)T + (1.38e7 - 8.74e7i)T^{2} \) |
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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
−14.04245444050493558276829677318, −13.00405793966240903981266763591, −11.50861855856148216033140151389, −10.36826474392586261554170664970, −9.340811587619776537719369803619, −7.940774406005685719877462671355, −6.92111596915952890709341787577, −4.79355987457551998603773826927, −3.36631682137723997093886158438, −2.66886786172017632840393619933,
1.63789834800289570585450479842, 3.28927602919412439702511029210, 4.54505135528942770672843423259, 6.54364998451320173007212489323, 7.920279917477206832139420800054, 8.524660363196380623640837900251, 9.856748866033942740624671188776, 11.84361452654671037134672333420, 12.71633009204899140661072884669, 13.43653897327448610467692815744
