| L(s) = 1 | + 0.656·2-s − 1.91·3-s − 1.56·4-s + 3.56·5-s − 1.25·6-s − 2.34·8-s + 0.656·9-s + 2.34·10-s − 11-s + 3·12-s + 5.91·13-s − 6.82·15-s + 1.59·16-s + 1.65·17-s + 0.431·18-s − 1.48·19-s − 5.59·20-s − 0.656·22-s + 3.34·23-s + 4.48·24-s + 7.73·25-s + 3.88·26-s + 4.48·27-s + 3.08·29-s − 4.48·30-s + 7.08·31-s + 5.73·32-s + ⋯ |
| L(s) = 1 | + 0.464·2-s − 1.10·3-s − 0.784·4-s + 1.59·5-s − 0.512·6-s − 0.828·8-s + 0.218·9-s + 0.741·10-s − 0.301·11-s + 0.866·12-s + 1.63·13-s − 1.76·15-s + 0.399·16-s + 0.401·17-s + 0.101·18-s − 0.339·19-s − 1.25·20-s − 0.139·22-s + 0.697·23-s + 0.914·24-s + 1.54·25-s + 0.761·26-s + 0.862·27-s + 0.571·29-s − 0.818·30-s + 1.27·31-s + 1.01·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.364756028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.364756028\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 - 0.656T + 2T^{2} \) |
| 3 | \( 1 + 1.91T + 3T^{2} \) |
| 5 | \( 1 - 3.56T + 5T^{2} \) |
| 13 | \( 1 - 5.91T + 13T^{2} \) |
| 17 | \( 1 - 1.65T + 17T^{2} \) |
| 19 | \( 1 + 1.48T + 19T^{2} \) |
| 23 | \( 1 - 3.34T + 23T^{2} \) |
| 29 | \( 1 - 3.08T + 29T^{2} \) |
| 31 | \( 1 - 7.08T + 31T^{2} \) |
| 37 | \( 1 + 4.51T + 37T^{2} \) |
| 41 | \( 1 - 1.28T + 41T^{2} \) |
| 43 | \( 1 - 1.59T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 - 9.22T + 53T^{2} \) |
| 59 | \( 1 + 8.85T + 59T^{2} \) |
| 61 | \( 1 - 6.68T + 61T^{2} \) |
| 67 | \( 1 + 9.82T + 67T^{2} \) |
| 71 | \( 1 + 8.61T + 71T^{2} \) |
| 73 | \( 1 + 4.56T + 73T^{2} \) |
| 79 | \( 1 + 6.39T + 79T^{2} \) |
| 83 | \( 1 + 0.167T + 83T^{2} \) |
| 89 | \( 1 - 2.56T + 89T^{2} \) |
| 97 | \( 1 + 9.73T + 97T^{2} \) |
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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
−10.66549390548653882719894748482, −10.17934295567213937515702100674, −9.088574935212982251568212430130, −8.437133285485138668386100460028, −6.62906459434319191193757294519, −5.90241123495673660455209322761, −5.47358938333047756661106659633, −4.46650659690360489552586332376, −2.96965503063315255397767175358, −1.12730522117960499003149354272,
1.12730522117960499003149354272, 2.96965503063315255397767175358, 4.46650659690360489552586332376, 5.47358938333047756661106659633, 5.90241123495673660455209322761, 6.62906459434319191193757294519, 8.437133285485138668386100460028, 9.088574935212982251568212430130, 10.17934295567213937515702100674, 10.66549390548653882719894748482
