| L(s) = 1 | + 1.73·2-s + 2.73·3-s + 0.999·4-s + 1.73·5-s + 4.73·6-s − 1.26·7-s − 1.73·8-s + 4.46·9-s + 2.99·10-s − 4.73·11-s + 2.73·12-s + 13-s − 2.19·14-s + 4.73·15-s − 5·16-s + 3.46·17-s + 7.73·18-s + 1.26·19-s + 1.73·20-s − 3.46·21-s − 8.19·22-s − 4.73·24-s − 2.00·25-s + 1.73·26-s + 3.99·27-s − 1.26·28-s − 0.464·29-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 1.57·3-s + 0.499·4-s + 0.774·5-s + 1.93·6-s − 0.479·7-s − 0.612·8-s + 1.48·9-s + 0.948·10-s − 1.42·11-s + 0.788·12-s + 0.277·13-s − 0.586·14-s + 1.22·15-s − 1.25·16-s + 0.840·17-s + 1.82·18-s + 0.290·19-s + 0.387·20-s − 0.755·21-s − 1.74·22-s − 0.965·24-s − 0.400·25-s + 0.339·26-s + 0.769·27-s − 0.239·28-s − 0.0861·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.982796199\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.982796199\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 - 2.73T + 3T^{2} \) |
| 5 | \( 1 - 1.73T + 5T^{2} \) |
| 7 | \( 1 + 1.26T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 29 | \( 1 + 0.464T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 + 6.46T + 41T^{2} \) |
| 43 | \( 1 - 12.9T + 43T^{2} \) |
| 47 | \( 1 - 10.7T + 47T^{2} \) |
| 53 | \( 1 - 4.26T + 53T^{2} \) |
| 59 | \( 1 + 2.19T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 - 9.46T + 67T^{2} \) |
| 71 | \( 1 - 7.26T + 71T^{2} \) |
| 73 | \( 1 - 5.39T + 73T^{2} \) |
| 79 | \( 1 - 9.46T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 7.73T + 89T^{2} \) |
| 97 | \( 1 - 1.73T + 97T^{2} \) |
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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
−10.72597168148021397438803394502, −9.783122889558220453073116628759, −9.151465360070440939193362755670, −8.170694743560841429013058542841, −7.25523333590635158362858372148, −5.94219244659903260630647815549, −5.18426843723476547671155101635, −3.84378524858900141202786043225, −3.04319042831323723777989619581, −2.21808488822614578032609200091,
2.21808488822614578032609200091, 3.04319042831323723777989619581, 3.84378524858900141202786043225, 5.18426843723476547671155101635, 5.94219244659903260630647815549, 7.25523333590635158362858372148, 8.170694743560841429013058542841, 9.151465360070440939193362755670, 9.783122889558220453073116628759, 10.72597168148021397438803394502
