L(s) = 1 | + (0.133 − 0.5i)3-s + (−1 + 2i)5-s + (−0.232 − 0.133i)7-s + (2.36 + 1.36i)9-s + (4.59 + 1.23i)11-s + (3 + 2i)13-s + (0.866 + 0.767i)15-s + (−2.86 + 0.767i)17-s + (−0.866 − 3.23i)19-s + (−0.0980 + 0.0980i)21-s + (−0.133 − 0.0358i)23-s + (−3 − 4i)25-s + (2.09 − 2.09i)27-s + (−1.03 + 0.598i)29-s + (2.26 + 2.26i)31-s + ⋯ |
L(s) = 1 | + (0.0773 − 0.288i)3-s + (−0.447 + 0.894i)5-s + (−0.0877 − 0.0506i)7-s + (0.788 + 0.455i)9-s + (1.38 + 0.371i)11-s + (0.832 + 0.554i)13-s + (0.223 + 0.198i)15-s + (−0.695 + 0.186i)17-s + (−0.198 − 0.741i)19-s + (−0.0214 + 0.0214i)21-s + (−0.0279 − 0.00748i)23-s + (−0.600 − 0.800i)25-s + (0.403 − 0.403i)27-s + (−0.192 + 0.111i)29-s + (0.407 + 0.407i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.26465 + 0.341702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.26465 + 0.341702i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (1 - 2i)T \) |
| 13 | \( 1 + (-3 - 2i)T \) |
good | 3 | \( 1 + (-0.133 + 0.5i)T + (-2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (0.232 + 0.133i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.59 - 1.23i)T + (9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 + (2.86 - 0.767i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.866 + 3.23i)T + (-16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 + (0.133 + 0.0358i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (1.03 - 0.598i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.26 - 2.26i)T + 31iT^{2} \) |
| 37 | \( 1 + (3.23 - 1.86i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.86 - 6.96i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.66 + 9.96i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 - 7.46iT - 47T^{2} \) |
| 53 | \( 1 + (8.46 + 8.46i)T + 53iT^{2} \) |
| 59 | \( 1 + (-13.7 + 3.69i)T + (51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.23 + 10.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.86 + 0.767i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 - 0.928T + 73T^{2} \) |
| 79 | \( 1 + 11.4iT - 79T^{2} \) |
| 83 | \( 1 + 3.46iT - 83T^{2} \) |
| 89 | \( 1 + (0.794 - 2.96i)T + (-77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (-1.23 + 2.13i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.90341736566227359578056214058, −11.21056646571931701769983121372, −10.27429989412980363624660931575, −9.182068357785622667546952044579, −8.107498813846741626312997285450, −6.74971256443723792136709754852, −6.64788167196521563928874840225, −4.58193133376093330025124441029, −3.57335143420865868677238473167, −1.83917472684112210544127837166,
1.25771307979135068966093774660, 3.64456328275796076045641318957, 4.35584187622343341792790672418, 5.81953960837875859378702042148, 6.90604809138374494331939667463, 8.252047313603479156648744273117, 9.000117031525289714409109734392, 9.835633949759491437321759689945, 11.06946078957426409823089273629, 11.96147327865989119037248429661