| L(s) = 1 | − 2.39·2-s + 3.73·4-s + 1.75·5-s + 4.73·7-s − 4.14·8-s − 4.19·10-s + 4.14·11-s + 3.46·13-s − 11.3·14-s + 2.46·16-s − 19-s + 6.54·20-s − 9.92·22-s − 6.54·23-s − 1.92·25-s − 8.29·26-s + 17.6·28-s + 5.42·29-s − 6.66·31-s + 2.39·32-s + 8.29·35-s − 6.19·37-s + 2.39·38-s − 7.26·40-s − 10.6·41-s + 0.732·43-s + 15.4·44-s + ⋯ |
| L(s) = 1 | − 1.69·2-s + 1.86·4-s + 0.783·5-s + 1.78·7-s − 1.46·8-s − 1.32·10-s + 1.25·11-s + 0.960·13-s − 3.02·14-s + 0.616·16-s − 0.229·19-s + 1.46·20-s − 2.11·22-s − 1.36·23-s − 0.385·25-s − 1.62·26-s + 3.33·28-s + 1.00·29-s − 1.19·31-s + 0.423·32-s + 1.40·35-s − 1.01·37-s + 0.388·38-s − 1.14·40-s − 1.66·41-s + 0.111·43-s + 2.33·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9794918866\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9794918866\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 2 | \( 1 + 2.39T + 2T^{2} \) |
| 5 | \( 1 - 1.75T + 5T^{2} \) |
| 7 | \( 1 - 4.73T + 7T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 6.54T + 23T^{2} \) |
| 29 | \( 1 - 5.42T + 29T^{2} \) |
| 31 | \( 1 + 6.66T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 + 10.6T + 41T^{2} \) |
| 43 | \( 1 - 0.732T + 43T^{2} \) |
| 47 | \( 1 - 7.18T + 47T^{2} \) |
| 53 | \( 1 - 7.18T + 53T^{2} \) |
| 59 | \( 1 - 3.03T + 59T^{2} \) |
| 61 | \( 1 - 3.19T + 61T^{2} \) |
| 67 | \( 1 + 2.26T + 67T^{2} \) |
| 71 | \( 1 + 6.07T + 71T^{2} \) |
| 73 | \( 1 + 8.12T + 73T^{2} \) |
| 79 | \( 1 + 9T + 79T^{2} \) |
| 83 | \( 1 - 11.9T + 83T^{2} \) |
| 89 | \( 1 + 15.9T + 89T^{2} \) |
| 97 | \( 1 - 9.66T + 97T^{2} \) |
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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
−10.62290445422939831614463523646, −10.01303800627984914761633810061, −8.820481336153256567277606198706, −8.586038623671998130948986290932, −7.58443213981732406632221566126, −6.58456994819020947333312622076, −5.57606700929463580808759657917, −4.09294135795672062501703509357, −1.97980266540319254504783919051, −1.38721538957833394241138147876,
1.38721538957833394241138147876, 1.97980266540319254504783919051, 4.09294135795672062501703509357, 5.57606700929463580808759657917, 6.58456994819020947333312622076, 7.58443213981732406632221566126, 8.586038623671998130948986290932, 8.820481336153256567277606198706, 10.01303800627984914761633810061, 10.62290445422939831614463523646
