| L(s) = 1 | + 3.69·2-s + (−11.8 − 20.4i)3-s − 18.3·4-s + (21.7 + 37.7i)5-s + (−43.6 − 75.6i)6-s + (−19.0 + 32.9i)7-s − 186.·8-s + (−158. + 273. i)9-s + (80.4 + 139. i)10-s − 444.·11-s + (216. + 375. i)12-s + (−340. + 588. i)13-s + (−70.3 + 121. i)14-s + (514. − 891. i)15-s − 99.9·16-s + (541. − 937. i)17-s + ⋯ |
| L(s) = 1 | + 0.653·2-s + (−0.758 − 1.31i)3-s − 0.573·4-s + (0.389 + 0.674i)5-s + (−0.495 − 0.857i)6-s + (−0.146 + 0.254i)7-s − 1.02·8-s + (−0.650 + 1.12i)9-s + (0.254 + 0.440i)10-s − 1.10·11-s + (0.434 + 0.753i)12-s + (−0.558 + 0.966i)13-s + (−0.0959 + 0.166i)14-s + (0.590 − 1.02i)15-s − 0.0976·16-s + (0.454 − 0.786i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.886 - 0.462i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0364151 + 0.148617i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0364151 + 0.148617i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.13e3 - 1.13e4i)T \) |
| good | 2 | \( 1 - 3.69T + 32T^{2} \) |
| 3 | \( 1 + (11.8 + 20.4i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (-21.7 - 37.7i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (19.0 - 32.9i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + 444.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (340. - 588. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-541. + 937. i)T + (-7.09e5 - 1.22e6i)T^{2} \) |
| 19 | \( 1 + (1.27e3 + 2.20e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-498. - 862. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (2.08e3 - 3.60e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (3.78e3 + 6.54e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (2.34e3 + 4.06e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + 6.61e3T + 1.15e8T^{2} \) |
| 47 | \( 1 + 7.39e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-3.27e3 - 5.67e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + 1.87e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (1.93e4 - 3.34e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.59e4 + 2.76e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.91e4 + 6.77e4i)T + (-9.02e8 - 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-2.84e4 + 4.93e4i)T + (-1.03e9 - 1.79e9i)T^{2} \) |
| 79 | \( 1 + (1.98e4 - 3.44e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-2.04e4 - 3.54e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-1.44e3 - 2.50e3i)T + (-2.79e9 + 4.83e9i)T^{2} \) |
| 97 | \( 1 + 9.66e4T + 8.58e9T^{2} \) |
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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
−13.89543266645139044194276594783, −13.11404978751871416956215145306, −12.23389575941453250812801033977, −11.03203344239481294526234371113, −9.295299666058713454024085975610, −7.41156690780182897498994093019, −6.26962313695697811506299953598, −5.03795153206826911744887144885, −2.52821136190692787932533063911, −0.07111515780253443552142099649,
3.69615495198956500821584130799, 5.05709325313784078198735488618, 5.67193428963493953096772009463, 8.399061359396433416449170998322, 9.882595243415778130963849162949, 10.53458393808417452213773666103, 12.36509861925256578185348444793, 13.10566163464860462595079320502, 14.63139626336246367234940769556, 15.51306045479766648747764441719
