| L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.293 + 1.66i)3-s + (0.766 − 0.642i)4-s + (−3.02 − 2.53i)5-s + (0.293 + 1.66i)6-s + (−1.16 − 2.01i)7-s + (0.500 − 0.866i)8-s + (0.132 + 0.0481i)9-s + (−3.70 − 1.34i)10-s + (0.5 − 0.866i)11-s + (0.845 + 1.46i)12-s + (−1.16 − 6.60i)13-s + (−1.78 − 1.49i)14-s + (5.10 − 4.28i)15-s + (0.173 − 0.984i)16-s + (6.72 − 2.44i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (−0.169 + 0.961i)3-s + (0.383 − 0.321i)4-s + (−1.35 − 1.13i)5-s + (0.119 + 0.679i)6-s + (−0.439 − 0.761i)7-s + (0.176 − 0.306i)8-s + (0.0441 + 0.0160i)9-s + (−1.17 − 0.426i)10-s + (0.150 − 0.261i)11-s + (0.244 + 0.422i)12-s + (−0.323 − 1.83i)13-s + (−0.476 − 0.399i)14-s + (1.31 − 1.10i)15-s + (0.0434 − 0.246i)16-s + (1.63 − 0.593i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0180 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0180 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.921030 - 0.904546i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.921030 - 0.904546i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 19 | \( 1 + (3.39 - 2.73i)T \) |
| good | 3 | \( 1 + (0.293 - 1.66i)T + (-2.81 - 1.02i)T^{2} \) |
| 5 | \( 1 + (3.02 + 2.53i)T + (0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (1.16 + 2.01i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (1.16 + 6.60i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-6.72 + 2.44i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (2.01 - 1.68i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.93 + 1.06i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (2.83 + 4.91i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.47T + 37T^{2} \) |
| 41 | \( 1 + (-0.00883 + 0.0500i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (2.09 + 1.75i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.51 - 0.550i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (4.62 - 3.88i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (1.79 - 0.653i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.57 + 1.32i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.01 - 1.46i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-10.7 - 9.04i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-2.42 + 13.7i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (-0.744 + 4.22i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.59 - 4.49i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.47 - 8.33i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-14.8 + 5.41i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.99550200020068962689618078416, −10.19647688273936313765035803098, −9.472334649815920779400391881987, −7.958957834983824790326274232406, −7.57323087651379801714646989036, −5.72448910007759558157531768024, −4.94922593202278596871597821025, −3.92662378719270839653564383383, −3.45038504493917013145693202753, −0.70018849644242229166484921008,
2.14792368294229779544931707891, 3.45703903204654611535043970606, 4.42328168755345547674379575890, 6.11104076073472425321072797000, 6.78656289462300982536214790121, 7.36874305178765869606522842851, 8.284122046620611462659208661391, 9.636537171279086897302300759785, 10.95173606287286416127080416101, 11.76621244156040511728494354206
