| L(s) = 1 | − 3.93i·2-s − 11.5·4-s + 7.90·5-s + 3.69i·7-s + 29.5i·8-s − 31.1i·10-s + 11.7·11-s + 12.4·13-s + 14.5·14-s + 70.3·16-s + (−12.3 − 11.6i)17-s − 9.34·19-s − 90.9·20-s − 46.1i·22-s + 34.2·23-s + ⋯ |
| L(s) = 1 | − 1.96i·2-s − 2.87·4-s + 1.58·5-s + 0.528i·7-s + 3.69i·8-s − 3.11i·10-s + 1.06·11-s + 0.960·13-s + 1.03·14-s + 4.39·16-s + (−0.729 − 0.684i)17-s − 0.491·19-s − 4.54·20-s − 2.09i·22-s + 1.49·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 459 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.684 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.026604626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.026604626\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + (12.3 + 11.6i)T \) |
| good | 2 | \( 1 + 3.93iT - 4T^{2} \) |
| 5 | \( 1 - 7.90T + 25T^{2} \) |
| 7 | \( 1 - 3.69iT - 49T^{2} \) |
| 11 | \( 1 - 11.7T + 121T^{2} \) |
| 13 | \( 1 - 12.4T + 169T^{2} \) |
| 19 | \( 1 + 9.34T + 361T^{2} \) |
| 23 | \( 1 - 34.2T + 529T^{2} \) |
| 29 | \( 1 - 9.56T + 841T^{2} \) |
| 31 | \( 1 + 31.3iT - 961T^{2} \) |
| 37 | \( 1 - 30.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 51.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 12.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 29.6iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 20.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 73.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 76.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 7.61T + 4.48e3T^{2} \) |
| 71 | \( 1 + 37.1T + 5.04e3T^{2} \) |
| 73 | \( 1 + 23.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 35.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 54.0iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 35.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 6.48iT - 9.40e3T^{2} \) |
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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
−10.62928401159828159387801943381, −9.685402627236237778871114194389, −9.115667173333569861806727626252, −8.596979504648632349771560091620, −6.45929258536552482842377188591, −5.43948276251129824992356033193, −4.42804981514953077806090273543, −3.09296721361693872699109143791, −2.11109173637790608421095509277, −1.14607106129469090819519666810,
1.24341806664090518978934313914, 3.76490808192011442827783389465, 4.87233342677728407895669408427, 5.87392869063403128897840021284, 6.53150380940738246891213713902, 7.09404606092704557536133262900, 8.721422175336616535963380991439, 8.847046804094726212501193654482, 9.917156971067037033590597420384, 10.75785117769875677557299873636
