L(s) = 1 | + (−1.41 − 1.41i)2-s + (−2.79 + 1.15i)3-s + (−0.0129 + 3.99i)4-s + (−1.86 + 4.49i)5-s + (5.58 + 2.32i)6-s + (0.944 + 6.93i)7-s + (5.68 − 5.62i)8-s + (0.104 − 0.104i)9-s + (8.99 − 3.70i)10-s + (2.63 − 6.36i)11-s + (−4.59 − 11.1i)12-s + (−3.81 − 9.20i)13-s + (8.49 − 11.1i)14-s − 14.7i·15-s + (−15.9 − 0.103i)16-s − 18.4·17-s + ⋯ |
L(s) = 1 | + (−0.705 − 0.708i)2-s + (−0.931 + 0.385i)3-s + (−0.00324 + 0.999i)4-s + (−0.372 + 0.898i)5-s + (0.930 + 0.387i)6-s + (0.134 + 0.990i)7-s + (0.710 − 0.703i)8-s + (0.0115 − 0.0115i)9-s + (0.899 − 0.370i)10-s + (0.239 − 0.578i)11-s + (−0.382 − 0.932i)12-s + (−0.293 − 0.708i)13-s + (0.606 − 0.795i)14-s − 0.980i·15-s + (−0.999 − 0.00649i)16-s − 1.08·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.00895650 - 0.0548803i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00895650 - 0.0548803i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.41 + 1.41i)T \) |
| 7 | \( 1 + (-0.944 - 6.93i)T \) |
good | 3 | \( 1 + (2.79 - 1.15i)T + (6.36 - 6.36i)T^{2} \) |
| 5 | \( 1 + (1.86 - 4.49i)T + (-17.6 - 17.6i)T^{2} \) |
| 11 | \( 1 + (-2.63 + 6.36i)T + (-85.5 - 85.5i)T^{2} \) |
| 13 | \( 1 + (3.81 + 9.20i)T + (-119. + 119. i)T^{2} \) |
| 17 | \( 1 + 18.4T + 289T^{2} \) |
| 19 | \( 1 + (-6.94 - 16.7i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (1.72 - 1.72i)T - 529iT^{2} \) |
| 29 | \( 1 + (17.1 + 41.4i)T + (-594. + 594. i)T^{2} \) |
| 31 | \( 1 + 2.69iT - 961T^{2} \) |
| 37 | \( 1 + (63.1 + 26.1i)T + (968. + 968. i)T^{2} \) |
| 41 | \( 1 + (-4.62 - 4.62i)T + 1.68e3iT^{2} \) |
| 43 | \( 1 + (-1.15 + 2.78i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + 9.80T + 2.20e3T^{2} \) |
| 53 | \( 1 + (-10.8 + 26.2i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-30.6 + 73.9i)T + (-2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-25.7 + 10.6i)T + (2.63e3 - 2.63e3i)T^{2} \) |
| 67 | \( 1 + (-29.2 - 70.5i)T + (-3.17e3 + 3.17e3i)T^{2} \) |
| 71 | \( 1 + (-33.8 - 33.8i)T + 5.04e3iT^{2} \) |
| 73 | \( 1 + (70.7 + 70.7i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 89.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (54.0 + 130. i)T + (-4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-72.3 + 72.3i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 - 108. iT - 9.40e3T^{2} \) |
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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
−11.99541798102874118286576155679, −11.45075709637212938424517273544, −10.79156564271557050734987430119, −9.938441471525064618878365446403, −8.746078923191707369343379225439, −7.78144796794242193576722753302, −6.45941013199546994347073154045, −5.30124268389589127372333039014, −3.69950140838623515525796217981, −2.39174041655241606371667287577,
0.04538633463181669397646998618, 1.36602519029325736261555170963, 4.44106454830355300145474049120, 5.23835711676191279052312116181, 6.78952782844003575254547137361, 7.09737776843279837935341993727, 8.549424363203081126508110010099, 9.312281360329363273227764748386, 10.58919631795234855009374731723, 11.37711234564205870683970248964