L(s) = 1 | + (−0.5 − 0.866i)2-s + (1.60 + 0.646i)3-s + (−0.499 + 0.866i)4-s + (−2.60 − 1.50i)5-s + (−0.243 − 1.71i)6-s + (−0.916 + 2.48i)7-s + 0.999·8-s + (2.16 + 2.07i)9-s + 3.00i·10-s − 4.03·11-s + (−1.36 + 1.06i)12-s + (−0.138 + 3.60i)13-s + (2.60 − 0.446i)14-s + (−3.21 − 4.09i)15-s + (−0.5 − 0.866i)16-s + (−1.67 + 2.89i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (0.927 + 0.373i)3-s + (−0.249 + 0.433i)4-s + (−1.16 − 0.672i)5-s + (−0.0992 − 0.700i)6-s + (−0.346 + 0.938i)7-s + 0.353·8-s + (0.721 + 0.692i)9-s + 0.950i·10-s − 1.21·11-s + (−0.393 + 0.308i)12-s + (−0.0382 + 0.999i)13-s + (0.696 − 0.119i)14-s + (−0.828 − 1.05i)15-s + (−0.125 − 0.216i)16-s + (−0.405 + 0.702i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.157 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.442621 + 0.518864i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.442621 + 0.518864i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (-1.60 - 0.646i)T \) |
| 7 | \( 1 + (0.916 - 2.48i)T \) |
| 13 | \( 1 + (0.138 - 3.60i)T \) |
good | 5 | \( 1 + (2.60 + 1.50i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + 4.03T + 11T^{2} \) |
| 17 | \( 1 + (1.67 - 2.89i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + 5.13T + 19T^{2} \) |
| 23 | \( 1 + (-2.14 + 1.24i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.74 - 2.16i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.95 - 5.12i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.22 - 1.28i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.29 + 3.63i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.23 + 3.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.77 - 3.33i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.50 - 4.90i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.75 - 1.59i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + 6.66iT - 61T^{2} \) |
| 67 | \( 1 - 8.14iT - 67T^{2} \) |
| 71 | \( 1 + (-0.436 - 0.756i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.78 - 11.7i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.50 + 14.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 12.6iT - 83T^{2} \) |
| 89 | \( 1 + (3.15 - 1.81i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.10 + 1.91i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.88136365318740102409041670675, −10.19243147086597637670502354287, −8.965215794730701367123805204656, −8.615441345977717365019220431386, −7.982917371820314052747231339957, −6.77154011984651917004231481415, −4.99121971768176254238093472435, −4.23208074152222887800100416420, −3.12473059739107629032765198287, −2.04063992580048511239539074789,
0.37955266339975604343230119946, 2.67344668980434208758673851294, 3.66212397524129910250659967538, 4.77524232073423187966723511302, 6.45876454483451390030756214552, 7.18915786183942546000779548430, 7.915445977276994719911557962683, 8.299551108775080504784327424129, 9.665592385351913173637294877278, 10.43361830123603329652937785185