| L(s) = 1 | + 2.95i·2-s + (6.98 + 4.03i)3-s + 7.24·4-s + (4.46 + 2.58i)5-s + (−11.9 + 20.6i)6-s + (37.3 − 21.5i)7-s + 68.7i·8-s + (−7.98 − 13.8i)9-s + (−7.63 + 13.2i)10-s − 174.·11-s + (50.6 + 29.2i)12-s + (139. + 242. i)13-s + (63.7 + 110. i)14-s + (20.8 + 36.0i)15-s − 87.5·16-s + (−238. − 413. i)17-s + ⋯ |
| L(s) = 1 | + 0.739i·2-s + (0.775 + 0.448i)3-s + 0.452·4-s + (0.178 + 0.103i)5-s + (−0.331 + 0.573i)6-s + (0.761 − 0.439i)7-s + 1.07i·8-s + (−0.0985 − 0.170i)9-s + (−0.0763 + 0.132i)10-s − 1.43·11-s + (0.351 + 0.202i)12-s + (0.827 + 1.43i)13-s + (0.325 + 0.563i)14-s + (0.0924 + 0.160i)15-s − 0.342·16-s + (−0.826 − 1.43i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.248 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.70466 + 1.32303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70466 + 1.32303i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.82e3 - 302. i)T \) |
| good | 2 | \( 1 - 2.95iT - 16T^{2} \) |
| 3 | \( 1 + (-6.98 - 4.03i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-4.46 - 2.58i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-37.3 + 21.5i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + 174.T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-139. - 242. i)T + (-1.42e4 + 2.47e4i)T^{2} \) |
| 17 | \( 1 + (238. + 413. i)T + (-4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (-85.0 - 49.1i)T + (6.51e4 + 1.12e5i)T^{2} \) |
| 23 | \( 1 + (-250. + 433. i)T + (-1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-849. + 490. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (394. - 682. i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + (-47.6 - 27.5i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + 1.79e3T + 2.82e6T^{2} \) |
| 47 | \( 1 - 545.T + 4.87e6T^{2} \) |
| 53 | \( 1 + (883. - 1.53e3i)T + (-3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + 4.42e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + (-2.51e3 + 1.45e3i)T + (6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-2.26e3 + 3.92e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (1.77e3 - 1.02e3i)T + (1.27e7 - 2.20e7i)T^{2} \) |
| 73 | \( 1 + (4.62e3 - 2.67e3i)T + (1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (1.84e3 + 3.18e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (2.11e3 - 3.66e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (-1.07e4 - 6.20e3i)T + (3.13e7 + 5.43e7i)T^{2} \) |
| 97 | \( 1 + 2.21e3T + 8.85e7T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.62146909821858003082681110155, −14.28265485582764857164401341307, −13.80362664127508908231163156189, −11.69705833609579764392207048976, −10.61048195853981198528279939285, −8.962484508354743798205317278392, −7.88486450878899719918738632782, −6.53852771933816866684159015488, −4.71769459374485749785467316706, −2.53096408141374734271490843553,
1.77713638445461002686671728548, 3.08075146158726261385149968417, 5.57906333951828837914277126365, 7.63159888929502287368800927378, 8.556764539571475356560402762455, 10.43065688516455051115173883503, 11.20556220522896659334173393031, 12.86192381153356357667638127266, 13.33718059669523869048263527610, 15.06739893695386835216267031712
