| L(s) = 1 | + (−0.327 + 0.945i)2-s + (−0.458 − 0.888i)3-s + (−0.786 − 0.618i)4-s + (−1.81 + 0.173i)5-s + (0.989 − 0.142i)6-s + (−1.72 + 2.00i)7-s + (0.841 − 0.540i)8-s + (−0.580 + 0.814i)9-s + (0.430 − 1.77i)10-s + (−2.76 + 0.956i)11-s + (−0.189 + 0.981i)12-s + (−1.18 − 4.04i)13-s + (−1.33 − 2.28i)14-s + (0.986 + 1.53i)15-s + (0.235 + 0.971i)16-s + (2.93 + 1.17i)17-s + ⋯ |
| L(s) = 1 | + (−0.231 + 0.668i)2-s + (−0.264 − 0.513i)3-s + (−0.393 − 0.309i)4-s + (−0.812 + 0.0775i)5-s + (0.404 − 0.0580i)6-s + (−0.652 + 0.757i)7-s + (0.297 − 0.191i)8-s + (−0.193 + 0.271i)9-s + (0.136 − 0.560i)10-s + (−0.832 + 0.288i)11-s + (−0.0546 + 0.283i)12-s + (−0.329 − 1.12i)13-s + (−0.355 − 0.611i)14-s + (0.254 + 0.396i)15-s + (0.0589 + 0.242i)16-s + (0.712 + 0.285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.765444 - 0.0322664i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.765444 - 0.0322664i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.327 - 0.945i)T \) |
| 3 | \( 1 + (0.458 + 0.888i)T \) |
| 7 | \( 1 + (1.72 - 2.00i)T \) |
| 23 | \( 1 + (-3.30 - 3.47i)T \) |
| good | 5 | \( 1 + (1.81 - 0.173i)T + (4.90 - 0.946i)T^{2} \) |
| 11 | \( 1 + (2.76 - 0.956i)T + (8.64 - 6.79i)T^{2} \) |
| 13 | \( 1 + (1.18 + 4.04i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (-2.93 - 1.17i)T + (12.3 + 11.7i)T^{2} \) |
| 19 | \( 1 + (-5.27 + 2.11i)T + (13.7 - 13.1i)T^{2} \) |
| 29 | \( 1 + (0.523 + 3.64i)T + (-27.8 + 8.17i)T^{2} \) |
| 31 | \( 1 + (-3.49 + 0.166i)T + (30.8 - 2.94i)T^{2} \) |
| 37 | \( 1 + (-0.0262 - 0.0187i)T + (12.1 + 34.9i)T^{2} \) |
| 41 | \( 1 + (3.07 + 1.40i)T + (26.8 + 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.83 + 7.53i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + (-6.45 + 3.72i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.81 - 2.94i)T + (-2.52 + 52.9i)T^{2} \) |
| 59 | \( 1 + (-3.40 - 0.826i)T + (52.4 + 27.0i)T^{2} \) |
| 61 | \( 1 + (-11.3 - 5.82i)T + (35.3 + 49.6i)T^{2} \) |
| 67 | \( 1 + (2.20 + 11.4i)T + (-62.2 + 24.9i)T^{2} \) |
| 71 | \( 1 + (-3.46 - 4.00i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-5.72 + 7.27i)T + (-17.2 - 70.9i)T^{2} \) |
| 79 | \( 1 + (-9.88 + 10.3i)T + (-3.75 - 78.9i)T^{2} \) |
| 83 | \( 1 + (0.933 + 2.04i)T + (-54.3 + 62.7i)T^{2} \) |
| 89 | \( 1 + (0.726 - 15.2i)T + (-88.5 - 8.45i)T^{2} \) |
| 97 | \( 1 + (4.32 - 9.48i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−9.922563236326344701333267299992, −9.070504961948389002160530202568, −7.956495900582948537343726167538, −7.63701571114809709802541217154, −6.77065579675475416182932409068, −5.54161940512620727431696553722, −5.28278253492688972937059784665, −3.63999744252610188710628214388, −2.58652509531174843474612800406, −0.58661635246176554324103306056,
0.859158474134220642494905249944, 2.81267212699521327492965150064, 3.68379679304105026752121740121, 4.48864133875643994351223402780, 5.46330362505386857493658132729, 6.81600134652950536651059539966, 7.60022820191400888655179245484, 8.447711891778846888326936652696, 9.556979325687081524035336695514, 9.964441306428360705578815293965
