L(s) = 1 | + (−0.370 − 0.642i)3-s + (1.24 − 2.15i)5-s + (0.670 − 2.55i)7-s + (1.22 − 2.12i)9-s + (2.66 + 4.61i)11-s + 4.45·13-s − 1.84·15-s + (2.61 + 4.52i)17-s + (−0.986 + 1.70i)19-s + (−1.89 + 0.518i)21-s + (3.48 − 6.04i)23-s + (−0.591 − 1.02i)25-s − 4.04·27-s + 3.94·29-s + (−2.01 − 3.49i)31-s + ⋯ |
L(s) = 1 | + (−0.214 − 0.370i)3-s + (0.556 − 0.963i)5-s + (0.253 − 0.967i)7-s + (0.408 − 0.707i)9-s + (0.803 + 1.39i)11-s + 1.23·13-s − 0.475·15-s + (0.633 + 1.09i)17-s + (−0.226 + 0.392i)19-s + (−0.412 + 0.113i)21-s + (0.727 − 1.25i)23-s + (−0.118 − 0.204i)25-s − 0.777·27-s + 0.733·29-s + (−0.362 − 0.628i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.010674503\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.010674503\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.670 + 2.55i)T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + (0.370 + 0.642i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1.24 + 2.15i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.66 - 4.61i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 4.45T + 13T^{2} \) |
| 17 | \( 1 + (-2.61 - 4.52i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.986 - 1.70i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.48 + 6.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.94T + 29T^{2} \) |
| 31 | \( 1 + (2.01 + 3.49i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.22 - 7.30i)T + (-18.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + (-2.33 + 4.04i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.07 + 1.86i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.53 + 2.65i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.01 + 10.4i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.81 - 13.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 0.549T + 71T^{2} \) |
| 73 | \( 1 + (-3.05 - 5.28i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (7.28 - 12.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.2T + 83T^{2} \) |
| 89 | \( 1 + (1.51 - 2.61i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−9.827826418265931061923171462559, −8.720603890045518959698444267186, −8.158096674472717962320649539691, −6.80531491904409395581910639788, −6.58795471382800259331097673015, −5.35239981825342419031847904187, −4.34398046546920460181826774337, −3.70863937769688041342573659203, −1.63453234207946964961414097746, −1.14001055403663323111040418619,
1.48002887520008870339216811026, 2.84288108496579770290509596045, 3.59100709130272045483980457347, 5.04633830000563024372652857093, 5.73534160734945298411973303588, 6.46977595592940875349837984155, 7.39998448947284564265074475994, 8.555384470260714228315457677412, 9.098097380237539191530949238285, 10.02416877454232256191372689628