| L(s) = 1 | + (−0.327 − 0.945i)2-s + (−0.458 + 0.888i)3-s + (−0.786 + 0.618i)4-s + (−1.17 − 0.111i)5-s + (0.989 + 0.142i)6-s + (1.80 + 1.93i)7-s + (0.841 + 0.540i)8-s + (−0.580 − 0.814i)9-s + (0.277 + 1.14i)10-s + (0.654 + 0.226i)11-s + (−0.189 − 0.981i)12-s + (−0.727 + 2.47i)13-s + (1.23 − 2.34i)14-s + (0.635 − 0.989i)15-s + (0.235 − 0.971i)16-s + (3.55 − 1.42i)17-s + ⋯ |
| L(s) = 1 | + (−0.231 − 0.668i)2-s + (−0.264 + 0.513i)3-s + (−0.393 + 0.309i)4-s + (−0.523 − 0.0500i)5-s + (0.404 + 0.0580i)6-s + (0.683 + 0.729i)7-s + (0.297 + 0.191i)8-s + (−0.193 − 0.271i)9-s + (0.0876 + 0.361i)10-s + (0.197 + 0.0682i)11-s + (−0.0546 − 0.283i)12-s + (−0.201 + 0.687i)13-s + (0.329 − 0.625i)14-s + (0.164 − 0.255i)15-s + (0.0589 − 0.242i)16-s + (0.863 − 0.345i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.434558 + 0.549313i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.434558 + 0.549313i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.80 - 1.93i)T \) |
| 23 | \( 1 + (-3.80 - 2.91i)T \) |
| good | 5 | \( 1 + (1.17 + 0.111i)T + (4.90 + 0.946i)T^{2} \) |
| 11 | \( 1 + (-0.654 - 0.226i)T + (8.64 + 6.79i)T^{2} \) |
| 13 | \( 1 + (0.727 - 2.47i)T + (-10.9 - 7.02i)T^{2} \) |
| 17 | \( 1 + (-3.55 + 1.42i)T + (12.3 - 11.7i)T^{2} \) |
| 19 | \( 1 + (6.95 + 2.78i)T + (13.7 + 13.1i)T^{2} \) |
| 29 | \( 1 + (1.02 - 7.14i)T + (-27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (6.04 + 0.288i)T + (30.8 + 2.94i)T^{2} \) |
| 37 | \( 1 + (5.16 - 3.67i)T + (12.1 - 34.9i)T^{2} \) |
| 41 | \( 1 + (-7.63 + 3.48i)T + (26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (-4.24 - 6.60i)T + (-17.8 + 39.1i)T^{2} \) |
| 47 | \( 1 + (5.53 + 3.19i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (7.46 - 7.83i)T + (-2.52 - 52.9i)T^{2} \) |
| 59 | \( 1 + (2.11 - 0.513i)T + (52.4 - 27.0i)T^{2} \) |
| 61 | \( 1 + (0.602 - 0.310i)T + (35.3 - 49.6i)T^{2} \) |
| 67 | \( 1 + (2.64 - 13.7i)T + (-62.2 - 24.9i)T^{2} \) |
| 71 | \( 1 + (3.89 - 4.49i)T + (-10.1 - 70.2i)T^{2} \) |
| 73 | \( 1 + (-5.29 - 6.73i)T + (-17.2 + 70.9i)T^{2} \) |
| 79 | \( 1 + (11.2 + 11.8i)T + (-3.75 + 78.9i)T^{2} \) |
| 83 | \( 1 + (-2.04 + 4.48i)T + (-54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-0.388 - 8.15i)T + (-88.5 + 8.45i)T^{2} \) |
| 97 | \( 1 + (-1.11 - 2.44i)T + (-63.5 + 73.3i)T^{2} \) |
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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
−10.41821828892600454370044791883, −9.253105272574463834463437093492, −8.947529208428155416150292570521, −7.940131919893297476703008075650, −7.00646392350189213602505186206, −5.71387017628060267668929319763, −4.81084095588464009052192723594, −4.04299523379254643996524721649, −2.88846332165583835966568735474, −1.59998335692542206630878742481,
0.37131204319453143456454463615, 1.84046342753762488601621112057, 3.64666138420558248593580485960, 4.55465750839058266031212805030, 5.63127178056954609036185876790, 6.43012302502685174183374555016, 7.48106861407341316870247296717, 7.87261596973841474229228924080, 8.597956862644977783958215508318, 9.784965505166258890201765926160
