| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.909 − 0.415i)3-s + (−0.959 + 0.281i)4-s + (2.43 − 2.81i)5-s + (0.281 − 0.959i)6-s + (−2.26 + 1.37i)7-s + (−0.415 − 0.909i)8-s + (0.654 + 0.755i)9-s + (3.12 + 2.01i)10-s + (−5.43 − 0.781i)11-s + (0.989 + 0.142i)12-s + (0.737 − 1.14i)13-s + (−1.68 − 2.04i)14-s + (−3.38 + 1.54i)15-s + (0.841 − 0.540i)16-s + (−0.428 − 0.125i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.525 − 0.239i)3-s + (−0.479 + 0.140i)4-s + (1.08 − 1.25i)5-s + (0.115 − 0.391i)6-s + (−0.854 + 0.519i)7-s + (−0.146 − 0.321i)8-s + (0.218 + 0.251i)9-s + (0.989 + 0.635i)10-s + (−1.63 − 0.235i)11-s + (0.285 + 0.0410i)12-s + (0.204 − 0.318i)13-s + (−0.449 − 0.546i)14-s + (−0.873 + 0.398i)15-s + (0.210 − 0.135i)16-s + (−0.103 − 0.0305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.746 + 0.665i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.121768 - 0.319371i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121768 - 0.319371i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (2.26 - 1.37i)T \) |
| 23 | \( 1 + (2.48 - 4.10i)T \) |
| good | 5 | \( 1 + (-2.43 + 2.81i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (5.43 + 0.781i)T + (10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.737 + 1.14i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (0.428 + 0.125i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.85 + 1.13i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (8.12 + 2.38i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (6.53 - 2.98i)T + (20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (5.76 - 4.99i)T + (5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (5.00 + 4.33i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.454 - 0.207i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 5.66iT - 47T^{2} \) |
| 53 | \( 1 + (4.15 + 6.45i)T + (-22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-3.71 + 5.77i)T + (-24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (1.01 + 2.22i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-2.31 + 0.332i)T + (64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.545 + 3.79i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.550 - 1.87i)T + (-61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (4.93 - 7.68i)T + (-32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (7.99 + 9.22i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (1.46 - 3.20i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (-6.92 + 7.99i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.628391029929449069230621950388, −8.865648297641733595107096719670, −8.016153270932321137240958184792, −7.08341828500537902019365575294, −5.88702313433867844632300544880, −5.50584155154569383436898007384, −4.99110987896505444992916053350, −3.33758545170771945063643154649, −1.86758047228000812328108854953, −0.15062945482256648204780721965,
1.97902055934559604696770187421, 2.95457003287685994605917583987, 3.84941855294321491876738653047, 5.29402980493746582711514301492, 5.87272280097846337540463938588, 6.87439077455098512386953134630, 7.60171067536550770664828369047, 9.161241031333655376999416931948, 9.850761627388118555011540770523, 10.54057151553796681689666684440
