| L(s) = 1 | + (−0.105 + 0.0383i)2-s + (−1.56 + 0.742i)3-s + (−1.52 + 1.27i)4-s + (0.729 − 4.13i)5-s + (0.136 − 0.138i)6-s + (0.565 − 0.979i)7-s + (0.223 − 0.386i)8-s + (1.89 − 2.32i)9-s + (0.0817 + 0.463i)10-s + 1.65·11-s + (1.43 − 3.12i)12-s + (−0.496 − 2.81i)13-s + (−0.0220 + 0.124i)14-s + (1.93 + 7.01i)15-s + (0.681 − 3.86i)16-s + (0.474 − 2.69i)17-s + ⋯ |
| L(s) = 1 | + (−0.0744 + 0.0271i)2-s + (−0.903 + 0.428i)3-s + (−0.761 + 0.638i)4-s + (0.326 − 1.84i)5-s + (0.0556 − 0.0564i)6-s + (0.213 − 0.370i)7-s + (0.0789 − 0.136i)8-s + (0.631 − 0.774i)9-s + (0.0258 + 0.146i)10-s + 0.499·11-s + (0.413 − 0.903i)12-s + (−0.137 − 0.781i)13-s + (−0.00588 + 0.0333i)14-s + (0.498 + 1.81i)15-s + (0.170 − 0.966i)16-s + (0.115 − 0.653i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.568178 - 0.396732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.568178 - 0.396732i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.56 - 0.742i)T \) |
| 19 | \( 1 + (4.08 - 1.50i)T \) |
| good | 2 | \( 1 + (0.105 - 0.0383i)T + (1.53 - 1.28i)T^{2} \) |
| 5 | \( 1 + (-0.729 + 4.13i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-0.565 + 0.979i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 1.65T + 11T^{2} \) |
| 13 | \( 1 + (0.496 + 2.81i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.474 + 2.69i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.50 + 2.10i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.65 - 3.91i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 - 1.44T + 31T^{2} \) |
| 37 | \( 1 + 1.65T + 37T^{2} \) |
| 41 | \( 1 + (-8.49 + 3.09i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-4.11 - 3.45i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-5.56 + 4.67i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + (-6.58 - 2.39i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (9.72 + 8.16i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.04 - 5.91i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-9.39 - 3.42i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (6.55 - 2.38i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (10.5 + 8.82i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (1.74 - 9.88i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (0.846 - 1.46i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-6.87 + 5.76i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-14.2 + 5.20i)T + (74.3 - 62.3i)T^{2} \) |
| 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
−12.64034041390925080736529866507, −11.86782373088246506170687569530, −10.47867869263833750673140076260, −9.336157464348674538469911226623, −8.779339645255591960141295268469, −7.52762579868888384054743103733, −5.74173979966721818311456254439, −4.81713576332275060566370144589, −4.05277106363420966767134296667, −0.794848880252705926447565909227,
2.05707002564562774842671701947, 4.17653484016551337549574610340, 5.77058904316446681527166589899, 6.42331712935915618004369498425, 7.47555320472768133895366601810, 9.130922064032409514833931053994, 10.23378713769658399141390944624, 10.91668235267079349119830128850, 11.67925247341075133111753820129, 13.05504292204572267101043436848
