| L(s) = 1 | + (0.939 − 0.342i)2-s + (0.173 − 0.984i)3-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.173 − 0.984i)6-s + (−2.11 − 3.66i)7-s + (0.500 − 0.866i)8-s + (−0.939 − 0.342i)9-s + (−0.939 − 0.342i)10-s + (1.06 − 1.83i)11-s + (−0.5 − 0.866i)12-s + (0.868 + 4.92i)13-s + (−3.23 − 2.71i)14-s + (−0.766 + 0.642i)15-s + (0.173 − 0.984i)16-s + (−3.10 + 1.13i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.100 − 0.568i)3-s + (0.383 − 0.321i)4-s + (−0.342 − 0.287i)5-s + (−0.0708 − 0.402i)6-s + (−0.798 − 1.38i)7-s + (0.176 − 0.306i)8-s + (−0.313 − 0.114i)9-s + (−0.297 − 0.108i)10-s + (0.319 − 0.553i)11-s + (−0.144 − 0.249i)12-s + (0.240 + 1.36i)13-s + (−0.865 − 0.726i)14-s + (−0.197 + 0.165i)15-s + (0.0434 − 0.246i)16-s + (−0.753 + 0.274i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.683 + 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.677745 - 1.56247i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.677745 - 1.56247i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (3.93 + 1.86i)T \) |
| good | 7 | \( 1 + (2.11 + 3.66i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.06 + 1.83i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.868 - 4.92i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (3.10 - 1.13i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-3.08 + 2.58i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-2.53 - 0.921i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.29 + 9.16i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + (-1.69 + 9.61i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-4.94 - 4.14i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-8.35 - 3.03i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (-6.05 + 5.07i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-8.92 + 3.24i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (9.88 - 8.29i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-2.57 - 0.936i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (7.04 + 5.90i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-0.0983 + 0.557i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (2.24 - 12.7i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-3.41 - 5.90i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.42 - 8.06i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-7.83 + 2.85i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.84902255246016056758500958761, −9.501690005049310974143550415026, −8.718150343978550184586141068778, −7.41861074769475882169770681919, −6.74290040525272102214664188926, −6.02848574787213229734622923271, −4.28403891394398948134451282158, −3.95384429530793840134949572664, −2.41222456189140128218598906236, −0.76895633756127334468015259787,
2.52752548512281226359857221160, 3.32104113619295812173599101957, 4.49857022739937508458948792373, 5.58245821246959647419293582659, 6.27858304704075911716180362957, 7.37939885519718893132696019036, 8.526874518634407868394258420394, 9.203484981053065293305638138545, 10.27916799835069419973180746843, 11.09108033155468011206735180577
