| L(s) = 1 | + (0.642 − 0.766i)2-s + (1.30 − 1.14i)3-s + (−0.173 − 0.984i)4-s + (0.582 + 2.15i)5-s + (−0.0408 − 1.73i)6-s + (−0.206 − 0.0364i)7-s + (−0.866 − 0.500i)8-s + (0.381 − 2.97i)9-s + (2.02 + 0.941i)10-s + (4.04 − 1.47i)11-s + (−1.35 − 1.08i)12-s + (−1.65 − 1.96i)13-s + (−0.160 + 0.134i)14-s + (3.22 + 2.14i)15-s + (−0.939 + 0.342i)16-s + (−0.534 + 0.308i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.750 − 0.660i)3-s + (−0.0868 − 0.492i)4-s + (0.260 + 0.965i)5-s + (−0.0166 − 0.706i)6-s + (−0.0780 − 0.0137i)7-s + (−0.306 − 0.176i)8-s + (0.127 − 0.991i)9-s + (0.641 + 0.297i)10-s + (1.22 − 0.444i)11-s + (−0.390 − 0.312i)12-s + (−0.457 − 0.545i)13-s + (−0.0429 + 0.0360i)14-s + (0.833 + 0.552i)15-s + (−0.234 + 0.0855i)16-s + (−0.129 + 0.0749i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.435 + 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.70508 - 1.06944i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.70508 - 1.06944i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-1.30 + 1.14i)T \) |
| 5 | \( 1 + (-0.582 - 2.15i)T \) |
| good | 7 | \( 1 + (0.206 + 0.0364i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-4.04 + 1.47i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.65 + 1.96i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.534 - 0.308i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.02 - 5.24i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.14 - 0.730i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (0.348 + 0.292i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.71 - 9.73i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (3.01 - 1.73i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.67 + 7.28i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.98 - 5.45i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-5.52 - 0.974i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 3.42iT - 53T^{2} \) |
| 59 | \( 1 + (2.84 + 1.03i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.575 + 3.26i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.34 + 9.94i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.880 - 1.52i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.16 - 4.13i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (9.21 + 7.73i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3.65 + 4.35i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (2.44 - 4.23i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.57 + 9.83i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.01833565949093084734637447008, −10.83550647579613448969811545821, −9.962881468749070728804921177525, −8.974785666869682805672227684418, −7.81053815240552530955333304746, −6.65893582490271991838094716474, −5.93301845051411791491152770602, −3.96072149565803256203380957435, −3.04734350374269491277955307203, −1.73864817959359801352686626511,
2.26289737425874065284468996735, 4.12397746877554831600659101065, 4.57575808653346932559358394144, 5.96350196770354447692676265467, 7.21378184520464775158080999321, 8.371956618265388436053325115922, 9.206434835520596655378619907168, 9.739211602892291588061407430884, 11.28312421918414228481634983351, 12.29115831665676021655249585202
