L(s) = 1 | + (0.623 + 0.781i)2-s + (0.808 + 2.06i)3-s + (−0.222 + 0.974i)4-s + (−0.0747 + 0.997i)5-s + (−1.10 + 1.91i)6-s + (1.63 + 2.83i)7-s + (−0.900 + 0.433i)8-s + (−1.39 + 1.29i)9-s + (−0.826 + 0.563i)10-s + (−0.960 − 4.20i)11-s + (−2.18 + 0.330i)12-s + (1.49 + 1.01i)13-s + (−1.19 + 3.04i)14-s + (−2.11 + 0.652i)15-s + (−0.900 − 0.433i)16-s + (−0.517 − 6.90i)17-s + ⋯ |
L(s) = 1 | + (0.440 + 0.552i)2-s + (0.467 + 1.18i)3-s + (−0.111 + 0.487i)4-s + (−0.0334 + 0.445i)5-s + (−0.451 + 0.782i)6-s + (0.618 + 1.07i)7-s + (−0.318 + 0.153i)8-s + (−0.464 + 0.431i)9-s + (−0.261 + 0.178i)10-s + (−0.289 − 1.26i)11-s + (−0.632 + 0.0952i)12-s + (0.413 + 0.281i)13-s + (−0.319 + 0.814i)14-s + (−0.546 + 0.168i)15-s + (−0.225 − 0.108i)16-s + (−0.125 − 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.716 - 0.698i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.771259 + 1.89586i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.771259 + 1.89586i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 5 | \( 1 + (0.0747 - 0.997i)T \) |
| 43 | \( 1 + (-6.03 - 2.55i)T \) |
good | 3 | \( 1 + (-0.808 - 2.06i)T + (-2.19 + 2.04i)T^{2} \) |
| 7 | \( 1 + (-1.63 - 2.83i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.960 + 4.20i)T + (-9.91 + 4.77i)T^{2} \) |
| 13 | \( 1 + (-1.49 - 1.01i)T + (4.74 + 12.1i)T^{2} \) |
| 17 | \( 1 + (0.517 + 6.90i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (0.235 + 0.218i)T + (1.41 + 18.9i)T^{2} \) |
| 23 | \( 1 + (-1.76 - 0.545i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (0.974 - 2.48i)T + (-21.2 - 19.7i)T^{2} \) |
| 31 | \( 1 + (1.49 - 0.224i)T + (29.6 - 9.13i)T^{2} \) |
| 37 | \( 1 + (2.51 - 4.35i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.32 + 4.16i)T + (-9.12 + 39.9i)T^{2} \) |
| 47 | \( 1 + (-0.126 + 0.555i)T + (-42.3 - 20.3i)T^{2} \) |
| 53 | \( 1 + (-3.12 + 2.12i)T + (19.3 - 49.3i)T^{2} \) |
| 59 | \( 1 + (-2.14 - 1.03i)T + (36.7 + 46.1i)T^{2} \) |
| 61 | \( 1 + (12.5 + 1.89i)T + (58.2 + 17.9i)T^{2} \) |
| 67 | \( 1 + (8.89 + 8.25i)T + (5.00 + 66.8i)T^{2} \) |
| 71 | \( 1 + (-15.7 + 4.84i)T + (58.6 - 39.9i)T^{2} \) |
| 73 | \( 1 + (-3.22 - 2.20i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (-1.68 - 2.91i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.35 - 11.0i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (-5.34 - 13.6i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (2.34 + 10.2i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−11.36175639622373902518236498596, −10.73430733165385762092139773356, −9.369807625902865146947920733070, −8.882104683354875127277714617182, −8.000929892290054028684984311727, −6.73126439503812328358387244519, −5.52594023024738489165918246870, −4.86372095235482093284713218080, −3.56088014255884584289872032322, −2.71838645911956437749333023009,
1.26035725845461324508180684848, 2.13949952639704056818589657614, 3.84541545918386677307289040617, 4.74987011343681440996883196373, 6.14191073228180319205534959641, 7.28621728252808217352664682598, 7.88808409750098193921899109514, 8.869571473745363784306151720956, 10.22184460510736431181611432323, 10.80019998435965284654927598034