| L(s) = 1 | + (−0.235 − 0.971i)2-s + (0.327 + 0.945i)3-s + (−0.888 + 0.458i)4-s + (1.60 − 0.640i)5-s + (0.841 − 0.540i)6-s + (2.54 − 0.725i)7-s + (0.654 + 0.755i)8-s + (−0.786 + 0.618i)9-s + (−0.999 − 1.40i)10-s + (0.979 − 4.03i)11-s + (−0.723 − 0.690i)12-s + (0.835 − 1.83i)13-s + (−1.30 − 2.30i)14-s + (1.12 + 1.30i)15-s + (0.580 − 0.814i)16-s + (0.205 + 4.30i)17-s + ⋯ |
| L(s) = 1 | + (−0.166 − 0.687i)2-s + (0.188 + 0.545i)3-s + (−0.444 + 0.229i)4-s + (0.715 − 0.286i)5-s + (0.343 − 0.220i)6-s + (0.961 − 0.274i)7-s + (0.231 + 0.267i)8-s + (−0.262 + 0.206i)9-s + (−0.316 − 0.444i)10-s + (0.295 − 1.21i)11-s + (−0.208 − 0.199i)12-s + (0.231 − 0.507i)13-s + (−0.348 − 0.615i)14-s + (0.291 + 0.336i)15-s + (0.145 − 0.203i)16-s + (0.0497 + 1.04i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.61189 - 0.946902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.61189 - 0.946902i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.235 + 0.971i)T \) |
| 3 | \( 1 + (-0.327 - 0.945i)T \) |
| 7 | \( 1 + (-2.54 + 0.725i)T \) |
| 23 | \( 1 + (4.18 - 2.33i)T \) |
| good | 5 | \( 1 + (-1.60 + 0.640i)T + (3.61 - 3.45i)T^{2} \) |
| 11 | \( 1 + (-0.979 + 4.03i)T + (-9.77 - 5.04i)T^{2} \) |
| 13 | \( 1 + (-0.835 + 1.83i)T + (-8.51 - 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.205 - 4.30i)T + (-16.9 + 1.61i)T^{2} \) |
| 19 | \( 1 + (-0.276 + 5.80i)T + (-18.9 - 1.80i)T^{2} \) |
| 29 | \( 1 + (-3.94 + 2.53i)T + (12.0 - 26.3i)T^{2} \) |
| 31 | \( 1 + (-2.20 + 0.424i)T + (28.7 - 11.5i)T^{2} \) |
| 37 | \( 1 + (-0.0568 + 0.0446i)T + (8.72 - 35.9i)T^{2} \) |
| 41 | \( 1 + (-0.0857 + 0.596i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.53 + 6.39i)T + (-6.11 - 42.5i)T^{2} \) |
| 47 | \( 1 + (-5.84 - 10.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (9.10 + 0.869i)T + (52.0 + 10.0i)T^{2} \) |
| 59 | \( 1 + (1.94 + 2.72i)T + (-19.2 + 55.7i)T^{2} \) |
| 61 | \( 1 + (-1.54 + 4.47i)T + (-47.9 - 37.7i)T^{2} \) |
| 67 | \( 1 + (8.62 - 8.22i)T + (3.18 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-12.7 - 3.74i)T + (59.7 + 38.3i)T^{2} \) |
| 73 | \( 1 + (-6.32 + 3.26i)T + (42.3 - 59.4i)T^{2} \) |
| 79 | \( 1 + (-10.6 + 1.01i)T + (77.5 - 14.9i)T^{2} \) |
| 83 | \( 1 + (0.326 + 2.27i)T + (-79.6 + 23.3i)T^{2} \) |
| 89 | \( 1 + (4.11 + 0.792i)T + (82.6 + 33.0i)T^{2} \) |
| 97 | \( 1 + (-0.730 + 5.08i)T + (-93.0 - 27.3i)T^{2} \) |
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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
−9.913853083594795346100353571955, −9.120571059416465054341176839584, −8.426300088937331634924368456959, −7.76000576980957807677588893036, −6.18457393216044018576147686553, −5.41240760146685105896880095318, −4.42011559116739513337015041486, −3.50458289975356918174495904060, −2.28979611430856905119725811115, −1.03301911390889747503873798371,
1.50262971936186071808409313047, 2.39903556408473352076254732686, 4.14367040096013882144623629128, 5.07872352944090196648497146102, 6.06635260515539511550958002387, 6.76858793798495955716646775920, 7.66935112018442055136497368714, 8.285650656503069441602585280537, 9.303444101357317992466734865364, 9.905367187115516428801753097660
