L(s) = 1 | + (0.654 − 0.755i)2-s + (−0.841 + 0.540i)3-s + (−0.142 − 0.989i)4-s + (0.477 + 1.04i)5-s + (−0.142 + 0.989i)6-s + (0.959 − 0.281i)7-s + (−0.841 − 0.540i)8-s + (0.415 − 0.909i)9-s + (1.10 + 0.324i)10-s + (−3.52 − 4.07i)11-s + (0.654 + 0.755i)12-s + (4.60 + 1.35i)13-s + (0.415 − 0.909i)14-s + (−0.967 − 0.621i)15-s + (−0.959 + 0.281i)16-s + (0.449 − 3.12i)17-s + ⋯ |
L(s) = 1 | + (0.463 − 0.534i)2-s + (−0.485 + 0.312i)3-s + (−0.0711 − 0.494i)4-s + (0.213 + 0.467i)5-s + (−0.0580 + 0.404i)6-s + (0.362 − 0.106i)7-s + (−0.297 − 0.191i)8-s + (0.138 − 0.303i)9-s + (0.348 + 0.102i)10-s + (−1.06 − 1.22i)11-s + (0.189 + 0.218i)12-s + (1.27 + 0.375i)13-s + (0.111 − 0.243i)14-s + (−0.249 − 0.160i)15-s + (−0.239 + 0.0704i)16-s + (0.108 − 0.757i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.50769 - 0.955567i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.50769 - 0.955567i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.654 + 0.755i)T \) |
| 3 | \( 1 + (0.841 - 0.540i)T \) |
| 7 | \( 1 + (-0.959 + 0.281i)T \) |
| 23 | \( 1 + (-2.69 - 3.97i)T \) |
good | 5 | \( 1 + (-0.477 - 1.04i)T + (-3.27 + 3.77i)T^{2} \) |
| 11 | \( 1 + (3.52 + 4.07i)T + (-1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-4.60 - 1.35i)T + (10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-0.449 + 3.12i)T + (-16.3 - 4.78i)T^{2} \) |
| 19 | \( 1 + (-0.222 - 1.54i)T + (-18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (-1.38 + 9.66i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (-1.65 - 1.06i)T + (12.8 + 28.1i)T^{2} \) |
| 37 | \( 1 + (-0.469 + 1.02i)T + (-24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (0.999 + 2.18i)T + (-26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-1.88 + 1.21i)T + (17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 + (-10.3 + 3.02i)T + (44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (3.97 + 1.16i)T + (49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (0.568 + 0.365i)T + (25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (6.14 - 7.09i)T + (-9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (8.76 - 10.1i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (1.92 + 13.3i)T + (-70.0 + 20.5i)T^{2} \) |
| 79 | \( 1 + (3.81 + 1.12i)T + (66.4 + 42.7i)T^{2} \) |
| 83 | \( 1 + (-5.52 + 12.0i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (10.2 - 6.57i)T + (36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-4.02 - 8.81i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.31597211883177298458853527925, −9.159865415718547536519190683223, −8.322371801048035436040462914107, −7.22163215558855318160379556905, −6.02464592753432122637764556018, −5.63469252077887169968713071678, −4.49942404123899002034361558111, −3.50187587214415339898472648541, −2.52926809366421058288101339777, −0.869469252165013056627077372907,
1.36296153943556972688101856843, 2.80600570230305618863955865560, 4.26993448765551831475740349611, 5.09041997513740015402693211442, 5.72013789348439518784229692146, 6.74715118501612846392265269136, 7.50609378397788252459447108406, 8.404654014625757869534294636029, 9.070734122824258027647671677384, 10.47220811385627204576042804922