| L(s) = 1 | + (−0.467 + 1.94i)2-s + (−1.79 + 2.40i)3-s + (−3.56 − 1.81i)4-s + (−1.89 − 3.28i)5-s + (−3.84 − 4.60i)6-s + (−5.70 + 9.88i)7-s + (5.20 − 6.07i)8-s + (−2.58 − 8.62i)9-s + (7.27 − 2.15i)10-s + (3.47 − 6.01i)11-s + (10.7 − 5.31i)12-s + (−14.6 + 8.48i)13-s + (−16.5 − 15.7i)14-s + (11.2 + 1.31i)15-s + (9.38 + 12.9i)16-s + 22.9i·17-s + ⋯ |
| L(s) = 1 | + (−0.233 + 0.972i)2-s + (−0.597 + 0.802i)3-s + (−0.890 − 0.454i)4-s + (−0.379 − 0.656i)5-s + (−0.640 − 0.768i)6-s + (−0.815 + 1.41i)7-s + (0.650 − 0.759i)8-s + (−0.287 − 0.957i)9-s + (0.727 − 0.215i)10-s + (0.315 − 0.546i)11-s + (0.896 − 0.442i)12-s + (−1.13 + 0.652i)13-s + (−1.18 − 1.12i)14-s + (0.753 + 0.0879i)15-s + (0.586 + 0.809i)16-s + 1.35i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.897 + 0.440i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0947541 - 0.408647i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0947541 - 0.408647i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.467 - 1.94i)T \) |
| 3 | \( 1 + (1.79 - 2.40i)T \) |
| good | 5 | \( 1 + (1.89 + 3.28i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 + (5.70 - 9.88i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.47 + 6.01i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (14.6 - 8.48i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 22.9iT - 289T^{2} \) |
| 19 | \( 1 - 21.7iT - 361T^{2} \) |
| 23 | \( 1 + (13.7 - 7.93i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-6.57 + 11.3i)T + (-420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + (-3.45 - 5.98i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 1.75iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-33.1 + 19.1i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-10.9 - 6.30i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (28.8 + 16.6i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + 1.96T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-10.5 - 18.2i)T + (-1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-48.0 - 27.7i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (75.4 - 43.5i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 38.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 31.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + (68.7 - 119. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-33.5 + 58.0i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 - 159. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-42.5 + 73.6i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.26226264632710262941474820388, −14.43480016242008752832401136887, −12.62717696867736470120644843453, −11.95642655870382162005016072560, −10.14141283253459642608360179777, −9.223059367582863279791346624470, −8.330941233747379899192894958778, −6.34703747558112989113669569708, −5.51710305814469693520495012908, −4.07238334394351321754033497092,
0.42044825799704099034309901789, 2.88087807730542712316851232444, 4.69595917729312938077387973288, 6.96429624029653534885456794985, 7.57279598658155116190214348690, 9.624094796327606091186307442625, 10.58083187849639152846784191463, 11.50564377909568358934918978430, 12.57851254394693400958896865463, 13.42463892303486366811431908229
