| L(s) = 1 | + (0.809 + 0.587i)2-s + (−0.00648 − 0.0199i)3-s + (0.309 + 0.951i)4-s + (2.20 + 0.370i)5-s + (0.00648 − 0.0199i)6-s − 4.68·7-s + (−0.309 + 0.951i)8-s + (2.42 − 1.76i)9-s + (1.56 + 1.59i)10-s + (2.39 + 1.74i)11-s + (0.0169 − 0.0123i)12-s + (5.59 − 4.06i)13-s + (−3.79 − 2.75i)14-s + (−0.00690 − 0.0463i)15-s + (−0.809 + 0.587i)16-s + (−0.180 + 0.554i)17-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (−0.00374 − 0.0115i)3-s + (0.154 + 0.475i)4-s + (0.986 + 0.165i)5-s + (0.00264 − 0.00814i)6-s − 1.77·7-s + (−0.109 + 0.336i)8-s + (0.808 − 0.587i)9-s + (0.495 + 0.504i)10-s + (0.723 + 0.525i)11-s + (0.00489 − 0.00355i)12-s + (1.55 − 1.12i)13-s + (−1.01 − 0.736i)14-s + (−0.00178 − 0.0119i)15-s + (−0.202 + 0.146i)16-s + (−0.0437 + 0.134i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 - 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.29946 + 0.969497i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.29946 + 0.969497i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-2.20 - 0.370i)T \) |
| 19 | \( 1 + (0.309 - 0.951i)T \) |
| good | 3 | \( 1 + (0.00648 + 0.0199i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 + 4.68T + 7T^{2} \) |
| 11 | \( 1 + (-2.39 - 1.74i)T + (3.39 + 10.4i)T^{2} \) |
| 13 | \( 1 + (-5.59 + 4.06i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.180 - 0.554i)T + (-13.7 - 9.99i)T^{2} \) |
| 23 | \( 1 + (-2.93 - 2.13i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.67 - 8.23i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.695 - 2.14i)T + (-25.0 - 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.07 + 0.783i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (8.70 - 6.32i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 1.07T + 43T^{2} \) |
| 47 | \( 1 + (1.82 + 5.63i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (2.04 + 6.28i)T + (-42.8 + 31.1i)T^{2} \) |
| 59 | \( 1 + (6.54 - 4.75i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-2.39 - 1.73i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + (-4.54 + 13.9i)T + (-54.2 - 39.3i)T^{2} \) |
| 71 | \( 1 + (1.21 + 3.74i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (3.84 + 2.79i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (0.878 + 2.70i)T + (-63.9 + 46.4i)T^{2} \) |
| 83 | \( 1 + (-3.38 + 10.4i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-2.91 - 2.11i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.424 - 1.30i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.09089726747702446293683337502, −9.347105050267783359015039130412, −8.643869152077509386749995099339, −7.17273682071278510328754445722, −6.47686554546842637973148046570, −6.16335350577543105117102432405, −5.04146115343976682995608775338, −3.54866840297792565187557905673, −3.23345589541480613673500162351, −1.39927090651309324008257348469,
1.21332749112887763195625128825, 2.48041858789003899143328383926, 3.61824727183320063021245541010, 4.41982071847148425655109584251, 5.76577385929825190138736403402, 6.44436541315715676963655902411, 6.85268818770349391502101577552, 8.602792214549187992733097094287, 9.354156571181872457944640856546, 9.900853833514686949901194202044
