| L(s) = 1 | + (0.288 − 0.344i)2-s + (0.312 + 1.77i)4-s + (−1.52 + 1.63i)5-s + (1.78 − 1.03i)7-s + (1.47 + 0.853i)8-s + (0.124 + 0.997i)10-s + (−1.15 + 1.99i)11-s + (−1.50 + 4.13i)13-s + (0.160 − 0.911i)14-s + (−2.65 + 0.967i)16-s + (3.51 − 4.19i)17-s + (−3.07 + 3.09i)19-s + (−3.37 − 2.18i)20-s + (0.354 + 0.973i)22-s + (−5.72 + 1.01i)23-s + ⋯ |
| L(s) = 1 | + (0.204 − 0.243i)2-s + (0.156 + 0.885i)4-s + (−0.680 + 0.732i)5-s + (0.674 − 0.389i)7-s + (0.522 + 0.301i)8-s + (0.0393 + 0.315i)10-s + (−0.347 + 0.602i)11-s + (−0.417 + 1.14i)13-s + (0.0429 − 0.243i)14-s + (−0.664 + 0.241i)16-s + (0.853 − 1.01i)17-s + (−0.704 + 0.709i)19-s + (−0.754 − 0.488i)20-s + (0.0755 + 0.207i)22-s + (−1.19 + 0.210i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.393 - 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.743462 + 1.12655i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.743462 + 1.12655i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.52 - 1.63i)T \) |
| 19 | \( 1 + (3.07 - 3.09i)T \) |
| good | 2 | \( 1 + (-0.288 + 0.344i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (-1.78 + 1.03i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.15 - 1.99i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.50 - 4.13i)T + (-9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-3.51 + 4.19i)T + (-2.95 - 16.7i)T^{2} \) |
| 23 | \( 1 + (5.72 - 1.01i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-4.21 + 3.53i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.378 + 0.656i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 6.22iT - 37T^{2} \) |
| 41 | \( 1 + (6.14 - 2.23i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-3.11 - 0.549i)T + (40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-4.08 - 4.87i)T + (-8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (3.79 - 0.668i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-1.95 - 1.63i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-1.72 - 9.78i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (1.09 + 1.30i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.71 - 9.70i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (3.49 + 9.60i)T + (-55.9 + 46.9i)T^{2} \) |
| 79 | \( 1 + (-2.26 + 0.824i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-6.11 + 3.53i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.19 - 0.798i)T + (68.1 + 57.2i)T^{2} \) |
| 97 | \( 1 + (-4.38 + 5.22i)T + (-16.8 - 95.5i)T^{2} \) |
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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
−10.51315570406864244410420297248, −9.820779434974791481610518386975, −8.495166010551520375463785773397, −7.70409382893306171898538687047, −7.32453128080956524062048482888, −6.30400822275740419939736617395, −4.66299490268917071811764705217, −4.17777802592654308145665761744, −3.03950044754033467326563601863, −1.98930459567770580548781215330,
0.60221133845811643939910654838, 2.02624103932145035207782719713, 3.61580649278219211004534712875, 4.82375586571122017175756841438, 5.39129937938555502666037437499, 6.21016908564739630954305349327, 7.49057839823260897593475283960, 8.224974669570525101974064736920, 8.856272427308454829968997432052, 10.12733327989047270066462889890
