| L(s) = 1 | + 6.57i·2-s − 27.2·4-s − 2.90i·5-s − 11.6·7-s − 74.3i·8-s + 19.1·10-s + 67.7i·11-s + 213.·13-s − 76.3i·14-s + 52.2·16-s + 235. i·17-s + 389.·19-s + 79.2i·20-s − 446.·22-s − 67.4i·23-s + ⋯ |
| L(s) = 1 | + 1.64i·2-s − 1.70·4-s − 0.116i·5-s − 0.236·7-s − 1.16i·8-s + 0.191·10-s + 0.560i·11-s + 1.26·13-s − 0.389i·14-s + 0.204·16-s + 0.816i·17-s + 1.07·19-s + 0.198i·20-s − 0.921·22-s − 0.127i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.557006255\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.557006255\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 453. iT \) |
| good | 2 | \( 1 - 6.57iT - 16T^{2} \) |
| 5 | \( 1 + 2.90iT - 625T^{2} \) |
| 7 | \( 1 + 11.6T + 2.40e3T^{2} \) |
| 11 | \( 1 - 67.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 213.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 235. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 389.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 67.4iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 920. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.11e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + 1.17e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.66e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.25e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.34e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 965. iT - 7.89e6T^{2} \) |
| 61 | \( 1 + 129.T + 1.38e7T^{2} \) |
| 67 | \( 1 + 2.85e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 1.72e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.60e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.38e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 1.70e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.47e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.67e4T + 8.85e7T^{2} \) |
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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
−10.55870244167625312692371110958, −9.556326536174276827192626214933, −8.629247834328728928452343300551, −8.080551530283917219273717913530, −6.96084686596572548423505246082, −6.39568691868541365418699288020, −5.41001481133898235679332702814, −4.53801819733416327859927613732, −3.28709970922965445863432073758, −1.28238637451362826505375701863,
0.45778388231031750509268036163, 1.41134499034895530472453384861, 2.82726522297068308050423248200, 3.46259936099921776519288483967, 4.58185786886766781867617142375, 5.78178495704362264520297679050, 6.97917896379640601618948855655, 8.344573503487346867975065977286, 9.108210108278747445032035376409, 9.945190051323140439343335955277
