| L(s) = 1 | + 0.266i·2-s + (−177. − 102. i)3-s + 1.02e3·4-s + (−1.11e3 − 642. i)5-s + (27.2 − 47.1i)6-s + (−3.82e3 + 2.20e3i)7-s + 544. i·8-s + (−8.61e3 − 1.49e4i)9-s + (170. − 295. i)10-s + 1.42e4·11-s + (−1.81e5 − 1.04e5i)12-s + (3.66e3 + 6.34e3i)13-s + (−587. − 1.01e3i)14-s + (1.31e5 + 2.27e5i)15-s + 1.04e6·16-s + (−9.20e5 − 1.59e6i)17-s + ⋯ |
| L(s) = 1 | + 0.00831i·2-s + (−0.728 − 0.420i)3-s + 0.999·4-s + (−0.355 − 0.205i)5-s + (0.00349 − 0.00606i)6-s + (−0.227 + 0.131i)7-s + 0.0166i·8-s + (−0.145 − 0.252i)9-s + (0.00170 − 0.00295i)10-s + 0.0885·11-s + (−0.728 − 0.420i)12-s + (0.00987 + 0.0170i)13-s + (−0.00109 − 0.00189i)14-s + (0.172 + 0.299i)15-s + 0.999·16-s + (−0.648 − 1.12i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.798 - 0.601i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.0198034 + 0.0592038i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0198034 + 0.0592038i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (7.80e7 - 1.24e8i)T \) |
| good | 2 | \( 1 - 0.266iT - 1.02e3T^{2} \) |
| 3 | \( 1 + (177. + 102. i)T + (2.95e4 + 5.11e4i)T^{2} \) |
| 5 | \( 1 + (1.11e3 + 642. i)T + (4.88e6 + 8.45e6i)T^{2} \) |
| 7 | \( 1 + (3.82e3 - 2.20e3i)T + (1.41e8 - 2.44e8i)T^{2} \) |
| 11 | \( 1 - 1.42e4T + 2.59e10T^{2} \) |
| 13 | \( 1 + (-3.66e3 - 6.34e3i)T + (-6.89e10 + 1.19e11i)T^{2} \) |
| 17 | \( 1 + (9.20e5 + 1.59e6i)T + (-1.00e12 + 1.74e12i)T^{2} \) |
| 19 | \( 1 + (-1.53e6 - 8.88e5i)T + (3.06e12 + 5.30e12i)T^{2} \) |
| 23 | \( 1 + (4.35e6 - 7.53e6i)T + (-2.07e13 - 3.58e13i)T^{2} \) |
| 29 | \( 1 + (2.56e7 - 1.48e7i)T + (2.10e14 - 3.64e14i)T^{2} \) |
| 31 | \( 1 + (1.70e7 - 2.94e7i)T + (-4.09e14 - 7.09e14i)T^{2} \) |
| 37 | \( 1 + (5.50e7 + 3.17e7i)T + (2.40e15 + 4.16e15i)T^{2} \) |
| 41 | \( 1 - 3.59e7T + 1.34e16T^{2} \) |
| 47 | \( 1 + 1.70e8T + 5.25e16T^{2} \) |
| 53 | \( 1 + (2.99e8 - 5.17e8i)T + (-8.74e16 - 1.51e17i)T^{2} \) |
| 59 | \( 1 + 5.02e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + (-1.11e9 + 6.41e8i)T + (3.56e17 - 6.17e17i)T^{2} \) |
| 67 | \( 1 + (-8.47e8 + 1.46e9i)T + (-9.11e17 - 1.57e18i)T^{2} \) |
| 71 | \( 1 + (-1.93e9 + 1.11e9i)T + (1.62e18 - 2.81e18i)T^{2} \) |
| 73 | \( 1 + (1.99e9 - 1.15e9i)T + (2.14e18 - 3.72e18i)T^{2} \) |
| 79 | \( 1 + (3.14e8 + 5.45e8i)T + (-4.73e18 + 8.19e18i)T^{2} \) |
| 83 | \( 1 + (-4.84e8 + 8.39e8i)T + (-7.75e18 - 1.34e19i)T^{2} \) |
| 89 | \( 1 + (5.24e9 + 3.02e9i)T + (1.55e19 + 2.70e19i)T^{2} \) |
| 97 | \( 1 - 9.46e9T + 7.37e19T^{2} \) |
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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
−14.25584377637059637103372337192, −12.63254939601361202048252722112, −11.79379916881748837897060896845, −11.05248627733947152826761609067, −9.410082535081017865185933137941, −7.63518760011175698130572632746, −6.59336737963617585024921155098, −5.43073323774942699460977909613, −3.31484832770748859085477563335, −1.55011173227598682933433846953,
0.02053036098640656422058068041, 2.08754952024542213164967828347, 3.81987954929593308889968622131, 5.57209791480549076358030969968, 6.72403250038712615295496299602, 8.080928606995119077830140179994, 10.01434097764958964032402121259, 11.05110189766707120361212405984, 11.69576755751828159872195153418, 13.10496059154287729124690528998
