L(s) = 1 | + (−2.11 + 1.22i)2-s − 3-s + (1.98 − 3.44i)4-s + (−3.36 − 1.94i)5-s + (2.11 − 1.22i)6-s + (−0.745 − 2.53i)7-s + 4.82i·8-s + 9-s + 9.50·10-s + 4.52i·11-s + (−1.98 + 3.44i)12-s + (−2.00 + 2.99i)13-s + (4.68 + 4.46i)14-s + (3.36 + 1.94i)15-s + (−1.92 − 3.32i)16-s + (3.16 − 5.47i)17-s + ⋯ |
L(s) = 1 | + (−1.49 + 0.864i)2-s − 0.577·3-s + (0.993 − 1.72i)4-s + (−1.50 − 0.869i)5-s + (0.864 − 0.498i)6-s + (−0.281 − 0.959i)7-s + 1.70i·8-s + 0.333·9-s + 3.00·10-s + 1.36i·11-s + (−0.573 + 0.993i)12-s + (−0.555 + 0.831i)13-s + (1.25 + 1.19i)14-s + (0.869 + 0.502i)15-s + (−0.480 − 0.831i)16-s + (0.766 − 1.32i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.134130 + 0.181142i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134130 + 0.181142i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + T \) |
| 7 | \( 1 + (0.745 + 2.53i)T \) |
| 13 | \( 1 + (2.00 - 2.99i)T \) |
good | 2 | \( 1 + (2.11 - 1.22i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (3.36 + 1.94i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 4.52iT - 11T^{2} \) |
| 17 | \( 1 + (-3.16 + 5.47i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 - 1.94iT - 19T^{2} \) |
| 23 | \( 1 + (-1.33 - 2.30i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.99 - 3.45i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.77 + 1.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.60 - 1.50i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-4.70 - 2.71i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.41 - 2.45i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.60 - 0.926i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.173 + 0.300i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.25 - 3.03i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 0.247iT - 67T^{2} \) |
| 71 | \( 1 + (-9.28 + 5.36i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.89 - 3.98i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.35 - 11.0i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 0.782iT - 83T^{2} \) |
| 89 | \( 1 + (10.6 - 6.15i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.4 - 6.59i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.93357923065405882894062271035, −11.11809197192881384203274841116, −9.891362264696437549861113328013, −9.395577068136964271919550904695, −8.102935574064667726226534158432, −7.25294230597753786793468638282, −7.01688049454320366278716290738, −5.16077485618137811046663908465, −4.13871556463092396655716493142, −1.06553322001924790239848807702,
0.37986308041237505079965718261, 2.73541961286316977010119004806, 3.64872336751449029195898595325, 5.77349633092896193509997265227, 7.08575695952902375739889847511, 8.101530111863656434833615547912, 8.606140872214966034283333825444, 9.991983819865329155464180945165, 10.80322146692350562938144902650, 11.33145894350814746541444079970