L(s) = 1 | − 2-s + (−0.646 − 1.60i)3-s + 4-s + (−2.87 + 1.66i)5-s + (0.646 + 1.60i)6-s + (−2.37 − 1.15i)7-s − 8-s + (−2.16 + 2.07i)9-s + (2.87 − 1.66i)10-s + (−0.741 − 1.28i)11-s + (−0.646 − 1.60i)12-s + (1.88 + 3.07i)13-s + (2.37 + 1.15i)14-s + (4.53 + 3.55i)15-s + 16-s + 5.63·17-s + ⋯ |
L(s) = 1 | − 0.707·2-s + (−0.373 − 0.927i)3-s + 0.5·4-s + (−1.28 + 0.743i)5-s + (0.263 + 0.655i)6-s + (−0.899 − 0.437i)7-s − 0.353·8-s + (−0.721 + 0.692i)9-s + (0.910 − 0.525i)10-s + (−0.223 − 0.387i)11-s + (−0.186 − 0.463i)12-s + (0.523 + 0.851i)13-s + (0.635 + 0.309i)14-s + (1.17 + 0.917i)15-s + 0.250·16-s + 1.36·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.543923 + 0.0264787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.543923 + 0.0264787i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + (0.646 + 1.60i)T \) |
| 7 | \( 1 + (2.37 + 1.15i)T \) |
| 13 | \( 1 + (-1.88 - 3.07i)T \) |
good | 5 | \( 1 + (2.87 - 1.66i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (0.741 + 1.28i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 - 5.63T + 17T^{2} \) |
| 19 | \( 1 + (-2.68 + 4.64i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 4.00iT - 23T^{2} \) |
| 29 | \( 1 + (0.127 + 0.0736i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.689 + 1.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 10.1iT - 37T^{2} \) |
| 41 | \( 1 + (0.728 + 0.420i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.56 + 7.90i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-8.41 + 4.85i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.6 - 6.15i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 0.151iT - 59T^{2} \) |
| 61 | \( 1 + (-6.32 - 3.65i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.61 - 4.97i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.25 - 3.90i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.99 + 3.44i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.75 - 3.03i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 1.72iT - 89T^{2} \) |
| 97 | \( 1 + (-7.27 - 12.5i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.89655402844537036080124793759, −10.09495096976206070560646792510, −8.881292535557988311020255084112, −7.923353313534430272870781833206, −7.17955007310641917208598850381, −6.74843236482887171288116413635, −5.57270216458691983230535392547, −3.74528163874011356533910919917, −2.84293193946886824054092014927, −0.891791541730053080981781407807,
0.60701741131924219119454624429, 3.14319372120692648716821000445, 3.91877901673458197705365888723, 5.25988114733245956401373553497, 6.06679851696047336067573621480, 7.50564540351412784331974743778, 8.262324639946814066438200561397, 9.054145113875294148928750509132, 9.962048803255778676695806647883, 10.54330615384629843943611276776