| 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
−11.37711234564205870683970248964, −10.58919631795234855009374731723, −9.312281360329363273227764748386, −8.549424363203081126508110010099, −7.09737776843279837935341993727, −6.78952782844003575254547137361, −5.23835711676191279052312116181, −4.44106454830355300145474049120, −1.36602519029325736261555170963, −0.04538633463181669397646998618,
2.39174041655241606371667287577, 3.69950140838623515525796217981, 5.30124268389589127372333039014, 6.45941013199546994347073154045, 7.78144796794242193576722753302, 8.746078923191707369343379225439, 9.938441471525064618878365446403, 10.79156564271557050734987430119, 11.45075709637212938424517273544, 11.99541798102874118286576155679
