| L(s) = 1 | + (0.425 + 0.533i)2-s + (2.86 + 0.431i)3-s + (7.01 − 30.7i)4-s + (−54.5 − 37.2i)5-s + (0.988 + 1.71i)6-s + (−110. + 191. i)7-s + (39.0 − 18.8i)8-s + (−224. − 69.1i)9-s + (−3.36 − 44.9i)10-s + (−26.2 − 115. i)11-s + (33.3 − 85.0i)12-s + (54.1 − 722. i)13-s + (−148. + 22.4i)14-s + (−140. − 130. i)15-s + (−882. − 425. i)16-s + (−1.44e3 + 982. i)17-s + ⋯ |
| L(s) = 1 | + (0.0751 + 0.0942i)2-s + (0.183 + 0.0277i)3-s + (0.219 − 0.960i)4-s + (−0.976 − 0.665i)5-s + (0.0112 + 0.0194i)6-s + (−0.851 + 1.47i)7-s + (0.215 − 0.103i)8-s + (−0.922 − 0.284i)9-s + (−0.0106 − 0.142i)10-s + (−0.0654 − 0.286i)11-s + (0.0669 − 0.170i)12-s + (0.0888 − 1.18i)13-s + (−0.203 + 0.0306i)14-s + (−0.160 − 0.149i)15-s + (−0.861 − 0.415i)16-s + (−1.20 + 0.824i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.137943 - 0.602914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.137943 - 0.602914i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-6.01e3 - 1.05e4i)T \) |
| good | 2 | \( 1 + (-0.425 - 0.533i)T + (-7.12 + 31.1i)T^{2} \) |
| 3 | \( 1 + (-2.86 - 0.431i)T + (232. + 71.6i)T^{2} \) |
| 5 | \( 1 + (54.5 + 37.2i)T + (1.14e3 + 2.90e3i)T^{2} \) |
| 7 | \( 1 + (110. - 191. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (26.2 + 115. i)T + (-1.45e5 + 6.98e4i)T^{2} \) |
| 13 | \( 1 + (-54.1 + 722. i)T + (-3.67e5 - 5.53e4i)T^{2} \) |
| 17 | \( 1 + (1.44e3 - 982. i)T + (5.18e5 - 1.32e6i)T^{2} \) |
| 19 | \( 1 + (-1.70e3 + 526. i)T + (2.04e6 - 1.39e6i)T^{2} \) |
| 23 | \( 1 + (-2.96e3 + 2.74e3i)T + (4.80e5 - 6.41e6i)T^{2} \) |
| 29 | \( 1 + (866. - 130. i)T + (1.95e7 - 6.04e6i)T^{2} \) |
| 31 | \( 1 + (819. - 2.08e3i)T + (-2.09e7 - 1.94e7i)T^{2} \) |
| 37 | \( 1 + (-3.09e3 - 5.35e3i)T + (-3.46e7 + 6.00e7i)T^{2} \) |
| 41 | \( 1 + (5.25e3 + 6.59e3i)T + (-2.57e7 + 1.12e8i)T^{2} \) |
| 47 | \( 1 + (72.4 - 317. i)T + (-2.06e8 - 9.95e7i)T^{2} \) |
| 53 | \( 1 + (1.99e3 + 2.66e4i)T + (-4.13e8 + 6.23e7i)T^{2} \) |
| 59 | \( 1 + (2.44e4 + 1.17e4i)T + (4.45e8 + 5.58e8i)T^{2} \) |
| 61 | \( 1 + (-7.17e3 - 1.82e4i)T + (-6.19e8 + 5.74e8i)T^{2} \) |
| 67 | \( 1 + (-1.46e4 + 4.50e3i)T + (1.11e9 - 7.60e8i)T^{2} \) |
| 71 | \( 1 + (5.15e4 + 4.78e4i)T + (1.34e8 + 1.79e9i)T^{2} \) |
| 73 | \( 1 + (-6.25e3 + 8.34e4i)T + (-2.04e9 - 3.08e8i)T^{2} \) |
| 79 | \( 1 + (3.74e4 - 6.48e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-5.08e4 - 7.66e3i)T + (3.76e9 + 1.16e9i)T^{2} \) |
| 89 | \( 1 + (1.99e4 + 3.00e3i)T + (5.33e9 + 1.64e9i)T^{2} \) |
| 97 | \( 1 + (-2.92e4 - 1.28e5i)T + (-7.73e9 + 3.72e9i)T^{2} \) |
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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
−14.90913251981860767804923322315, −13.16976178890138300616000495045, −12.07902997194545110228643455745, −10.94419108535209391937531791946, −9.245654570661015693934534003278, −8.406279314016647041560248999499, −6.31963777484891215412980608951, −5.18272039304841821129910552359, −2.91764062415215793104779161826, −0.30504733019274941402599185179,
3.04759697764092172784317028250, 4.10975814413079872076658342355, 7.00632294197147628293612576440, 7.50160547594559104929559514757, 9.218498830011292063074324794409, 11.05665243704284154689420946209, 11.63011214622102554318034585761, 13.26710312641343927273346005867, 14.02999867931654178155697021516, 15.64582709435160847135210846535
