| L(s) = 1 | + 1.41·2-s + 2.00·4-s + (−4.99 + 0.0147i)5-s − 5.91i·7-s + 2.82·8-s + (−7.07 + 0.0209i)10-s + 3.80i·11-s + 22.2i·13-s − 8.37i·14-s + 4.00·16-s + 1.20·17-s + 29.3·19-s + (−9.99 + 0.0295i)20-s + 5.38i·22-s + 16.1·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.500·4-s + (−0.999 + 0.00295i)5-s − 0.845i·7-s + 0.353·8-s + (−0.707 + 0.00209i)10-s + 0.346i·11-s + 1.70i·13-s − 0.597i·14-s + 0.250·16-s + 0.0708·17-s + 1.54·19-s + (−0.499 + 0.00147i)20-s + 0.244i·22-s + 0.701·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.00295i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.00295i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.552791099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.552791099\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (4.99 - 0.0147i)T \) |
| good | 7 | \( 1 + 5.91iT - 49T^{2} \) |
| 11 | \( 1 - 3.80iT - 121T^{2} \) |
| 13 | \( 1 - 22.2iT - 169T^{2} \) |
| 17 | \( 1 - 1.20T + 289T^{2} \) |
| 19 | \( 1 - 29.3T + 361T^{2} \) |
| 23 | \( 1 - 16.1T + 529T^{2} \) |
| 29 | \( 1 + 45.2iT - 841T^{2} \) |
| 31 | \( 1 - 43.7T + 961T^{2} \) |
| 37 | \( 1 + 48.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 2.72iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 19.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 24.2T + 2.20e3T^{2} \) |
| 53 | \( 1 - 10.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 19.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 18.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 83.1iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 105. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 94.2T + 6.24e3T^{2} \) |
| 83 | \( 1 + 94.6T + 6.88e3T^{2} \) |
| 89 | \( 1 - 132. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 119. iT - 9.40e3T^{2} \) |
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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
−10.14714226535071070608447058200, −9.273057235912532419977247193343, −8.127838734484653236333556126224, −7.18541781630392780442741126681, −6.85843842044949208335079272245, −5.46597609166028704212619683794, −4.27644934919602540298121654151, −3.99473995467465167354437793845, −2.62836848724595131444294552140, −0.998923519646784621021397128214,
0.954430519702534568979354425109, 2.99147753544138762418141587521, 3.28056296901390097734449091019, 4.84404641251095686075177158375, 5.38753761807425273832902694294, 6.46077159037452199592854042007, 7.55575868829248460281873987759, 8.153310762173477691451728859784, 9.085345718909684954703571230777, 10.28645220344159825330188021135
