| 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 + (−1.43 + 2.49i)7-s + (0.500 + 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.939 + 0.342i)10-s + (2.14 + 3.71i)11-s + (0.5 − 0.866i)12-s + (−0.205 + 1.16i)13-s + (−2.20 + 1.85i)14-s + (0.766 + 0.642i)15-s + (0.173 + 0.984i)16-s + (1.32 + 0.482i)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 + (−0.544 + 0.942i)7-s + (0.176 + 0.306i)8-s + (−0.313 + 0.114i)9-s + (−0.297 + 0.108i)10-s + (0.646 + 1.12i)11-s + (0.144 − 0.249i)12-s + (−0.0570 + 0.323i)13-s + (−0.589 + 0.494i)14-s + (0.197 + 0.165i)15-s + (0.0434 + 0.246i)16-s + (0.321 + 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45475 + 1.04112i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45475 + 1.04112i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.5 - 2.59i)T \) |
| good | 7 | \( 1 + (1.43 - 2.49i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.14 - 3.71i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (0.205 - 1.16i)T + (-12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.32 - 0.482i)T + (13.0 + 10.9i)T^{2} \) |
| 23 | \( 1 + (0.390 + 0.327i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (3.26 - 1.18i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-3.29 + 5.71i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 2.29T + 37T^{2} \) |
| 41 | \( 1 + (-0.886 - 5.02i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.60 + 6.38i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (4.23 - 1.54i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (8.36 + 7.02i)T + (9.20 + 52.1i)T^{2} \) |
| 59 | \( 1 + (6.68 + 2.43i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-10.3 - 8.67i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (5.15 - 1.87i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-5.10 + 4.28i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.20 - 6.86i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (3.05 + 17.3i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (1.02 - 1.76i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.435 - 2.47i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.1 - 3.67i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.30212739616402516678866332652, −9.963103082610439754379926272624, −9.163603190439678622960075736041, −7.970852050736411715269340484651, −7.18237874633260083827241293222, −6.35049594003712179553055042349, −5.54468552280826147815457451191, −4.30714204472339485370212287771, −3.13772938463925988846464321131, −1.92770443811309515266417563416,
0.873647871330714978105554528877, 3.12423741796904091830245310372, 3.76173044058227277670842854520, 4.79903077711401749353940232714, 5.81106139982521118793544399893, 6.79375864508195585767004342567, 7.82255403097037779970128814746, 9.004213248695764950692927766677, 9.819442393738761910465360033168, 10.75477018934934439373116962676
