| L(s) = 1 | − 1.73·2-s − 3-s + 0.999·4-s + 0.732·5-s + 1.73·6-s − 2·7-s + 1.73·8-s + 9-s − 1.26·10-s − 11-s − 0.999·12-s − 13-s + 3.46·14-s − 0.732·15-s − 5·16-s + 6.73·17-s − 1.73·18-s − 0.535·19-s + 0.732·20-s + 2·21-s + 1.73·22-s − 2·23-s − 1.73·24-s − 4.46·25-s + 1.73·26-s − 27-s − 1.99·28-s + ⋯ |
| L(s) = 1 | − 1.22·2-s − 0.577·3-s + 0.499·4-s + 0.327·5-s + 0.707·6-s − 0.755·7-s + 0.612·8-s + 0.333·9-s − 0.400·10-s − 0.301·11-s − 0.288·12-s − 0.277·13-s + 0.925·14-s − 0.189·15-s − 1.25·16-s + 1.63·17-s − 0.408·18-s − 0.122·19-s + 0.163·20-s + 0.436·21-s + 0.369·22-s − 0.417·23-s − 0.353·24-s − 0.892·25-s + 0.339·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 1.73T + 2T^{2} \) |
| 5 | \( 1 - 0.732T + 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 17 | \( 1 - 6.73T + 17T^{2} \) |
| 19 | \( 1 + 0.535T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 + 4.19T + 29T^{2} \) |
| 31 | \( 1 + 8.19T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 5.46T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 + 1.46T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 13.4T + 59T^{2} \) |
| 61 | \( 1 - 4.92T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 6.92T + 71T^{2} \) |
| 73 | \( 1 - 2.92T + 73T^{2} \) |
| 79 | \( 1 + 3.66T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 + 15.6T + 89T^{2} \) |
| 97 | \( 1 - 8.92T + 97T^{2} \) |
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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
−10.31850574761818891989063494270, −9.887918448489030988404690435313, −9.140383791686041291828294967045, −7.943656972539248090553778634377, −7.24340841132030452266532747571, −6.08334615241733373529407954872, −5.11423171313381413795979310471, −3.55474922112396448198929899788, −1.73227011454809870487556010975, 0,
1.73227011454809870487556010975, 3.55474922112396448198929899788, 5.11423171313381413795979310471, 6.08334615241733373529407954872, 7.24340841132030452266532747571, 7.943656972539248090553778634377, 9.140383791686041291828294967045, 9.887918448489030988404690435313, 10.31850574761818891989063494270
