| L(s) = 1 | + (1.36 + 0.366i)2-s + (2.95 − 0.538i)3-s + (1.73 + i)4-s + (−2.52 − 4.31i)5-s + (4.22 + 0.344i)6-s + (0.692 − 6.96i)7-s + (1.99 + 2i)8-s + (8.42 − 3.17i)9-s + (−1.86 − 6.82i)10-s + (−3.24 − 1.87i)11-s + (5.65 + 2.01i)12-s + (5.63 + 5.63i)13-s + (3.49 − 9.26i)14-s + (−9.76 − 11.3i)15-s + (1.99 + 3.46i)16-s + (4.24 − 1.13i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.983 − 0.179i)3-s + (0.433 + 0.250i)4-s + (−0.504 − 0.863i)5-s + (0.704 + 0.0574i)6-s + (0.0989 − 0.995i)7-s + (0.249 + 0.250i)8-s + (0.935 − 0.353i)9-s + (−0.186 − 0.682i)10-s + (−0.294 − 0.170i)11-s + (0.470 + 0.168i)12-s + (0.433 + 0.433i)13-s + (0.249 − 0.661i)14-s + (−0.651 − 0.758i)15-s + (0.124 + 0.216i)16-s + (0.249 − 0.0669i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 210 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.856 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.79081 - 0.775610i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.79081 - 0.775610i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.36 - 0.366i)T \) |
| 3 | \( 1 + (-2.95 + 0.538i)T \) |
| 5 | \( 1 + (2.52 + 4.31i)T \) |
| 7 | \( 1 + (-0.692 + 6.96i)T \) |
| good | 11 | \( 1 + (3.24 + 1.87i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-5.63 - 5.63i)T + 169iT^{2} \) |
| 17 | \( 1 + (-4.24 + 1.13i)T + (250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-8.60 - 14.8i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (9.56 - 35.7i)T + (-458. - 264.5i)T^{2} \) |
| 29 | \( 1 + 29.5T + 841T^{2} \) |
| 31 | \( 1 + (11.0 + 6.36i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (6.31 - 23.5i)T + (-1.18e3 - 684.5i)T^{2} \) |
| 41 | \( 1 - 26.2T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-38.5 + 38.5i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-7.31 + 27.3i)T + (-1.91e3 - 1.10e3i)T^{2} \) |
| 53 | \( 1 + (7.09 - 1.90i)T + (2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-68.7 - 39.6i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (47.6 - 27.5i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (79.7 - 21.3i)T + (3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 + 86.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-9.35 + 2.50i)T + (4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (134. - 77.7i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (68.5 - 68.5i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + (-94.3 + 54.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.8 - 17.8i)T - 9.40e3iT^{2} \) |
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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
−12.29813931908921159022671075498, −11.33748813785584045193413240159, −10.00774314120481818850915430493, −8.913999732518221335449984854821, −7.78774012704807788377062203060, −7.30446570129221610777425818897, −5.62797754643315204523998297330, −4.19716449418947714968887262555, −3.52271519607962340148854568045, −1.49083237832537466622743015001,
2.34565963032255165055361128981, 3.21298159155872285783263206748, 4.47438889993343935838033005240, 5.91300285879636232242924992788, 7.20788603437689779309669032164, 8.157609941851585763934356638334, 9.267124075312139084003810983725, 10.43921270613741001823107442315, 11.25731234740360885229081301774, 12.41384302848403788781245204446
