| L(s) = 1 | + (−3.87 + 2.81i)2-s + (4.62 − 14.2i)4-s + (−7.51 − 8.27i)5-s + 0.140·7-s + (10.3 + 31.7i)8-s + (52.4 + 10.9i)10-s + (38.3 − 27.8i)11-s + (−27.7 − 20.1i)13-s + (−0.544 + 0.395i)14-s + (−32.6 − 23.7i)16-s + (−5.10 − 15.7i)17-s + (28.1 + 86.5i)19-s + (−152. + 68.7i)20-s + (−70.1 + 215. i)22-s + (−130. + 94.9i)23-s + ⋯ |
| L(s) = 1 | + (−1.37 + 0.996i)2-s + (0.578 − 1.78i)4-s + (−0.672 − 0.740i)5-s + 0.00758·7-s + (0.456 + 1.40i)8-s + (1.65 + 0.344i)10-s + (1.05 − 0.762i)11-s + (−0.592 − 0.430i)13-s + (−0.0104 + 0.00755i)14-s + (−0.510 − 0.371i)16-s + (−0.0728 − 0.224i)17-s + (0.339 + 1.04i)19-s + (−1.70 + 0.769i)20-s + (−0.679 + 2.09i)22-s + (−1.18 + 0.861i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.00332682 - 0.0191388i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00332682 - 0.0191388i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (7.51 + 8.27i)T \) |
| good | 2 | \( 1 + (3.87 - 2.81i)T + (2.47 - 7.60i)T^{2} \) |
| 7 | \( 1 - 0.140T + 343T^{2} \) |
| 11 | \( 1 + (-38.3 + 27.8i)T + (411. - 1.26e3i)T^{2} \) |
| 13 | \( 1 + (27.7 + 20.1i)T + (678. + 2.08e3i)T^{2} \) |
| 17 | \( 1 + (5.10 + 15.7i)T + (-3.97e3 + 2.88e3i)T^{2} \) |
| 19 | \( 1 + (-28.1 - 86.5i)T + (-5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 + (130. - 94.9i)T + (3.75e3 - 1.15e4i)T^{2} \) |
| 29 | \( 1 + (-3.81 + 11.7i)T + (-1.97e4 - 1.43e4i)T^{2} \) |
| 31 | \( 1 + (102. + 316. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-227. - 165. i)T + (1.56e4 + 4.81e4i)T^{2} \) |
| 41 | \( 1 + (-101. - 73.8i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 + 529.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (23.1 - 71.3i)T + (-8.39e4 - 6.10e4i)T^{2} \) |
| 53 | \( 1 + (-77.0 + 237. i)T + (-1.20e5 - 8.75e4i)T^{2} \) |
| 59 | \( 1 + (217. + 157. i)T + (6.34e4 + 1.95e5i)T^{2} \) |
| 61 | \( 1 + (299. - 217. i)T + (7.01e4 - 2.15e5i)T^{2} \) |
| 67 | \( 1 + (54.5 + 167. i)T + (-2.43e5 + 1.76e5i)T^{2} \) |
| 71 | \( 1 + (223. - 686. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (-1.97 + 1.43i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (-187. + 577. i)T + (-3.98e5 - 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-392. - 1.20e3i)T + (-4.62e5 + 3.36e5i)T^{2} \) |
| 89 | \( 1 + (1.08e3 - 786. i)T + (2.17e5 - 6.70e5i)T^{2} \) |
| 97 | \( 1 + (442. - 1.36e3i)T + (-7.38e5 - 5.36e5i)T^{2} \) |
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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
−12.02532627120057523377350668669, −11.28987163385723749991885956453, −9.842787693415593142263770840755, −9.334030948457511595523195229176, −8.105549801411428617979824561461, −7.81903202439014698921619032648, −6.45223974494327831867028237429, −5.47651797854211137742150526382, −3.83739431546221899111396448780, −1.31860040635760425805962211242,
0.01374858260384652639103468902, 1.80156988442806237723715747345, 3.08499613041272746640955831001, 4.39099576639866814633975552676, 6.68081539539821263091498933743, 7.43339604902360118879892288468, 8.541529563100823331158052267463, 9.446225284777569646151796318336, 10.28270937552165089585024760370, 11.15177779426583436105437989233
