L(s) = 1 | + 2-s + (−1.18 − 1.26i)3-s + 4-s + (−0.686 + 2.12i)5-s + (−1.18 − 1.26i)6-s + 8-s + (−0.186 + 2.99i)9-s + (−0.686 + 2.12i)10-s − 4.25i·11-s + (−1.18 − 1.26i)12-s − 2·13-s + (3.5 − 1.65i)15-s + 16-s + 6.63i·17-s + (−0.186 + 2.99i)18-s + 3.46i·19-s + ⋯ |
L(s) = 1 | + 0.707·2-s + (−0.684 − 0.728i)3-s + 0.5·4-s + (−0.306 + 0.951i)5-s + (−0.484 − 0.515i)6-s + 0.353·8-s + (−0.0620 + 0.998i)9-s + (−0.216 + 0.672i)10-s − 1.28i·11-s + (−0.342 − 0.364i)12-s − 0.554·13-s + (0.903 − 0.428i)15-s + 0.250·16-s + 1.60i·17-s + (−0.0438 + 0.705i)18-s + 0.794i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.402 - 0.915i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.537382467\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.537382467\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.18 + 1.26i)T \) |
| 5 | \( 1 + (0.686 - 2.12i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + 4.25iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 6.63iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 - 4.37T + 23T^{2} \) |
| 29 | \( 1 - 3.31iT - 29T^{2} \) |
| 31 | \( 1 + 2.37iT - 31T^{2} \) |
| 37 | \( 1 - 11.6iT - 37T^{2} \) |
| 41 | \( 1 + 1.62T + 41T^{2} \) |
| 43 | \( 1 - 11.0iT - 43T^{2} \) |
| 47 | \( 1 - 1.87iT - 47T^{2} \) |
| 53 | \( 1 - 1.37T + 53T^{2} \) |
| 59 | \( 1 - 4.11T + 59T^{2} \) |
| 61 | \( 1 - 2.81iT - 61T^{2} \) |
| 67 | \( 1 - 7.57iT - 67T^{2} \) |
| 71 | \( 1 + 1.87iT - 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 + 8.11T + 79T^{2} \) |
| 83 | \( 1 - 1.43iT - 83T^{2} \) |
| 89 | \( 1 - 4.37T + 89T^{2} \) |
| 97 | \( 1 - 2.11T + 97T^{2} \) |
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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
−10.08326934573937713189916577472, −8.473763271138326662810594780775, −7.889790269603460793904709676197, −7.00323964868704676420015981155, −6.24501269294601135994408077377, −5.81657513244818157842941549000, −4.69774300794860612462565641912, −3.55375140120151152226146911297, −2.72568324518726942333394489658, −1.39510433385679228118722824199,
0.54523352781917476416152661981, 2.28457894640732996837154428910, 3.60673749271220186937471130670, 4.65231129068081108763746567165, 4.90311536913926839657572163902, 5.67759253219960676876459194717, 7.03991958905032637923632690870, 7.33374257436126633645589140907, 8.839701500623199849888993018413, 9.412838374922180467637632870229