| L(s) = 1 | + (−0.104 + 0.994i)2-s + (0.257 + 1.71i)3-s + (−0.978 − 0.207i)4-s + (−0.789 + 1.77i)5-s + (−1.73 + 0.0772i)6-s + (−2.28 + 1.32i)7-s + (0.309 − 0.951i)8-s + (−2.86 + 0.882i)9-s + (−1.68 − 0.970i)10-s + (−3.21 − 0.795i)11-s + (0.104 − 1.72i)12-s + (3.98 − 5.48i)13-s + (−1.07 − 2.41i)14-s + (−3.24 − 0.895i)15-s + (0.913 + 0.406i)16-s + (0.459 + 4.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.0739 + 0.703i)2-s + (0.148 + 0.988i)3-s + (−0.489 − 0.103i)4-s + (−0.353 + 0.792i)5-s + (−0.706 + 0.0315i)6-s + (−0.865 + 0.501i)7-s + (0.109 − 0.336i)8-s + (−0.955 + 0.294i)9-s + (−0.531 − 0.306i)10-s + (−0.970 − 0.239i)11-s + (0.0300 − 0.499i)12-s + (1.10 − 1.52i)13-s + (−0.288 − 0.645i)14-s + (−0.836 − 0.231i)15-s + (0.228 + 0.101i)16-s + (0.111 + 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.680 + 0.733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.680 + 0.733i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.259765 - 0.595427i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.259765 - 0.595427i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.104 - 0.994i)T \) |
| 3 | \( 1 + (-0.257 - 1.71i)T \) |
| 7 | \( 1 + (2.28 - 1.32i)T \) |
| 11 | \( 1 + (3.21 + 0.795i)T \) |
| good | 5 | \( 1 + (0.789 - 1.77i)T + (-3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (-3.98 + 5.48i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.459 - 4.36i)T + (-16.6 + 3.53i)T^{2} \) |
| 19 | \( 1 + (0.0977 + 0.459i)T + (-17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.703 + 0.406i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.927 - 2.85i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (7.66 - 3.41i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (6.60 - 7.33i)T + (-3.86 - 36.7i)T^{2} \) |
| 41 | \( 1 + (1.04 - 3.22i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 3.53iT - 43T^{2} \) |
| 47 | \( 1 + (1.94 + 9.13i)T + (-42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (1.70 + 3.82i)T + (-35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (-0.627 + 2.95i)T + (-53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (2.08 - 4.67i)T + (-40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (1.96 - 3.39i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.37 - 11.5i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (1.63 - 7.69i)T + (-66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (7.45 + 0.784i)T + (77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (4.29 - 3.12i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (11.9 - 6.90i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.56 - 1.13i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.22413765712515318531615147318, −10.49113888902936744781627104403, −9.999842650207702628534071305482, −8.636763029601600309010425400722, −8.271165897259989236420989106361, −6.96146137656172821643412008009, −5.86886620030134439816528002790, −5.24321055259720632804020184111, −3.56306011807877871577256608718, −3.08327963322230132663652241800,
0.39450032853226392315205583213, 1.89682733691347993035494292397, 3.28936070087436256841130951994, 4.42943328776105829780200809689, 5.76620778866500471764367049174, 6.95255037855708777232746077942, 7.75670824397282240046625110085, 8.890419391045110358575204470543, 9.348301208793282061012769873431, 10.68727726844236823760929239417
