| L(s) = 1 | + 3-s − 1.23·5-s + 2·7-s + 9-s − 1.23·11-s + 4.47·13-s − 1.23·15-s + 2.47·17-s + 3.23·19-s + 2·21-s + 23-s − 3.47·25-s + 27-s + 0.472·29-s + 10.4·31-s − 1.23·33-s − 2.47·35-s − 5.70·37-s + 4.47·39-s − 2·41-s + 0.763·43-s − 1.23·45-s + 8.94·47-s − 3·49-s + 2.47·51-s − 10.1·53-s + 1.52·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.552·5-s + 0.755·7-s + 0.333·9-s − 0.372·11-s + 1.24·13-s − 0.319·15-s + 0.599·17-s + 0.742·19-s + 0.436·21-s + 0.208·23-s − 0.694·25-s + 0.192·27-s + 0.0876·29-s + 1.88·31-s − 0.215·33-s − 0.417·35-s − 0.938·37-s + 0.716·39-s − 0.312·41-s + 0.116·43-s − 0.184·45-s + 1.30·47-s − 0.428·49-s + 0.346·51-s − 1.39·53-s + 0.206·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.819998646\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.819998646\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 1.23T + 5T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 - 2.47T + 17T^{2} \) |
| 19 | \( 1 - 3.23T + 19T^{2} \) |
| 29 | \( 1 - 0.472T + 29T^{2} \) |
| 31 | \( 1 - 10.4T + 31T^{2} \) |
| 37 | \( 1 + 5.70T + 37T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 - 0.763T + 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 + 10.1T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 - 1.70T + 61T^{2} \) |
| 67 | \( 1 + 5.70T + 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 4.47T + 73T^{2} \) |
| 79 | \( 1 - 14T + 79T^{2} \) |
| 83 | \( 1 + 6.76T + 83T^{2} \) |
| 89 | \( 1 - 0.944T + 89T^{2} \) |
| 97 | \( 1 + 4.47T + 97T^{2} \) |
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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
−10.80643068248448183545082342217, −9.923226024300271008707521892243, −8.820499236897661591862804253466, −8.095978193253137536512105470538, −7.51352871046068967894625126645, −6.23946313881954693312138465764, −5.06492443751225047616724074665, −3.99083789638915331981761180477, −2.97557145258673576731571604553, −1.36993016678430846634347929361,
1.36993016678430846634347929361, 2.97557145258673576731571604553, 3.99083789638915331981761180477, 5.06492443751225047616724074665, 6.23946313881954693312138465764, 7.51352871046068967894625126645, 8.095978193253137536512105470538, 8.820499236897661591862804253466, 9.923226024300271008707521892243, 10.80643068248448183545082342217
