| L(s) = 1 | − 3.07i·2-s − 5.44·4-s + 6.03·5-s − 9.30·7-s + 4.42i·8-s − 18.5i·10-s + 12.6i·11-s + 16.7i·13-s + 28.5i·14-s − 8.16·16-s − 14.0·17-s − 37.4·19-s − 32.8·20-s + 38.9·22-s − 27.8i·23-s + ⋯ |
| L(s) = 1 | − 1.53i·2-s − 1.36·4-s + 1.20·5-s − 1.32·7-s + 0.553i·8-s − 1.85i·10-s + 1.15i·11-s + 1.29i·13-s + 2.04i·14-s − 0.510·16-s − 0.828·17-s − 1.96·19-s − 1.64·20-s + 1.76·22-s − 1.20i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2667326910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2667326910\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + (-28.6 + 51.5i)T \) |
| good | 2 | \( 1 + 3.07iT - 4T^{2} \) |
| 5 | \( 1 - 6.03T + 25T^{2} \) |
| 7 | \( 1 + 9.30T + 49T^{2} \) |
| 11 | \( 1 - 12.6iT - 121T^{2} \) |
| 13 | \( 1 - 16.7iT - 169T^{2} \) |
| 17 | \( 1 + 14.0T + 289T^{2} \) |
| 19 | \( 1 + 37.4T + 361T^{2} \) |
| 23 | \( 1 + 27.8iT - 529T^{2} \) |
| 29 | \( 1 + 39.1T + 841T^{2} \) |
| 31 | \( 1 - 0.257iT - 961T^{2} \) |
| 37 | \( 1 - 54.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 11.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 7.36iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 28.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 47.3T + 2.80e3T^{2} \) |
| 61 | \( 1 + 84.0iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 81.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 6.96T + 5.04e3T^{2} \) |
| 73 | \( 1 - 131. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 51.9T + 6.24e3T^{2} \) |
| 83 | \( 1 + 16.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 69.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 110. iT - 9.40e3T^{2} \) |
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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
−10.62595652043474383279378273789, −9.969614069024994683100947681363, −9.428558121199909517647829210793, −8.762463955989256775312155496718, −6.68666995963008475695392011158, −6.45540452923357213802257716437, −4.71475299583515182381593233005, −3.86415680756822837796361322371, −2.35371267379855427759319337417, −1.95373190106741676559675829857,
0.092986541396735981711289372799, 2.45416505101892804787591122677, 3.88207031840135081358146427383, 5.58735310537959674539035023455, 5.86077238095547638892325767757, 6.59660487852900562610013555679, 7.60412931562472641521458884963, 8.753084821203045898913858947419, 9.224114926621270947815210290018, 10.26947258182969210816866904692
