L(s) = 1 | + (1.38 + 7.87i)2-s + (−9.71 + 8.15i)3-s + (−60.1 + 21.8i)4-s + (424. + 154. i)5-s + (−77.7 − 65.2i)6-s + (847. − 1.46e3i)7-s + (−256 − 443. i)8-s + (−351. + 1.99e3i)9-s + (−627. + 3.55e3i)10-s + (−355. − 615. i)11-s + (405. − 703. i)12-s + (7.81e3 + 6.55e3i)13-s + (1.27e4 + 4.63e3i)14-s + (−5.38e3 + 1.95e3i)15-s + (3.13e3 − 2.63e3i)16-s + (3.78e3 + 2.14e4i)17-s + ⋯ |
L(s) = 1 | + (0.122 + 0.696i)2-s + (−0.207 + 0.174i)3-s + (−0.469 + 0.171i)4-s + (1.51 + 0.552i)5-s + (−0.146 − 0.123i)6-s + (0.934 − 1.61i)7-s + (−0.176 − 0.306i)8-s + (−0.160 + 0.912i)9-s + (−0.198 + 1.12i)10-s + (−0.0804 − 0.139i)11-s + (0.0678 − 0.117i)12-s + (0.986 + 0.827i)13-s + (1.24 + 0.451i)14-s + (−0.411 + 0.149i)15-s + (0.191 − 0.160i)16-s + (0.186 + 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.226 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(1.82425 + 1.44826i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82425 + 1.44826i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.38 - 7.87i)T \) |
| 19 | \( 1 + (2.87e4 - 8.23e3i)T \) |
good | 3 | \( 1 + (9.71 - 8.15i)T + (379. - 2.15e3i)T^{2} \) |
| 5 | \( 1 + (-424. - 154. i)T + (5.98e4 + 5.02e4i)T^{2} \) |
| 7 | \( 1 + (-847. + 1.46e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (355. + 615. i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (-7.81e3 - 6.55e3i)T + (1.08e7 + 6.17e7i)T^{2} \) |
| 17 | \( 1 + (-3.78e3 - 2.14e4i)T + (-3.85e8 + 1.40e8i)T^{2} \) |
| 23 | \( 1 + (-4.30e4 + 1.56e4i)T + (2.60e9 - 2.18e9i)T^{2} \) |
| 29 | \( 1 + (-9.27e3 + 5.26e4i)T + (-1.62e10 - 5.89e9i)T^{2} \) |
| 31 | \( 1 + (3.25e4 - 5.64e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 2.62e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.53e5 + 1.29e5i)T + (3.38e10 - 1.91e11i)T^{2} \) |
| 43 | \( 1 + (-4.78e5 - 1.74e5i)T + (2.08e11 + 1.74e11i)T^{2} \) |
| 47 | \( 1 + (-1.34e5 + 7.61e5i)T + (-4.76e11 - 1.73e11i)T^{2} \) |
| 53 | \( 1 + (-1.20e6 + 4.37e5i)T + (8.99e11 - 7.55e11i)T^{2} \) |
| 59 | \( 1 + (-2.93e5 - 1.66e6i)T + (-2.33e12 + 8.51e11i)T^{2} \) |
| 61 | \( 1 + (1.49e6 - 5.44e5i)T + (2.40e12 - 2.02e12i)T^{2} \) |
| 67 | \( 1 + (2.20e5 - 1.24e6i)T + (-5.69e12 - 2.07e12i)T^{2} \) |
| 71 | \( 1 + (4.78e6 + 1.74e6i)T + (6.96e12 + 5.84e12i)T^{2} \) |
| 73 | \( 1 + (-4.18e6 + 3.51e6i)T + (1.91e12 - 1.08e13i)T^{2} \) |
| 79 | \( 1 + (4.40e6 - 3.69e6i)T + (3.33e12 - 1.89e13i)T^{2} \) |
| 83 | \( 1 + (-1.52e6 + 2.63e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (6.28e6 + 5.27e6i)T + (7.68e12 + 4.35e13i)T^{2} \) |
| 97 | \( 1 + (-7.67e5 - 4.35e6i)T + (-7.59e13 + 2.76e13i)T^{2} \) |
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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
−14.71697381169923526474857191011, −13.86282675066518232021140931384, −13.34444990802731519986266928039, −10.85812410162937530790385256145, −10.36027720748348530583766159414, −8.524117338138853453006846244373, −7.02516782766253445958584989266, −5.78438257410530742072529353212, −4.26407262410028146390297605570, −1.65001154429354452057133269588,
1.23306902637769674842907025264, 2.58007030736730046971866724821, 5.18433288861283468406509252385, 5.98320707083274159335892007403, 8.706629610280457024061700262059, 9.362506457518808146448979119876, 10.98708987413533018967003861092, 12.22448188437513644935454700285, 13.06324844791065563123925448167, 14.34680666413218240672040378510