| L(s) = 1 | − 3.23·3-s + 2·5-s + 3.23·7-s + 7.47·9-s + 3.23·11-s − 4.47·13-s − 6.47·15-s + 17-s + 2.47·19-s − 10.4·21-s + 3.23·23-s − 25-s − 14.4·27-s + 2·29-s − 3.23·31-s − 10.4·33-s + 6.47·35-s + 6.94·37-s + 14.4·39-s + 2·41-s − 10.4·43-s + 14.9·45-s − 4.94·47-s + 3.47·49-s − 3.23·51-s − 2·53-s + 6.47·55-s + ⋯ |
| L(s) = 1 | − 1.86·3-s + 0.894·5-s + 1.22·7-s + 2.49·9-s + 0.975·11-s − 1.24·13-s − 1.67·15-s + 0.242·17-s + 0.567·19-s − 2.28·21-s + 0.674·23-s − 0.200·25-s − 2.78·27-s + 0.371·29-s − 0.581·31-s − 1.82·33-s + 1.09·35-s + 1.14·37-s + 2.31·39-s + 0.312·41-s − 1.59·43-s + 2.22·45-s − 0.721·47-s + 0.496·49-s − 0.453·51-s − 0.274·53-s + 0.872·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8274875531\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8274875531\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 + 3.23T + 3T^{2} \) |
| 5 | \( 1 - 2T + 5T^{2} \) |
| 7 | \( 1 - 3.23T + 7T^{2} \) |
| 11 | \( 1 - 3.23T + 11T^{2} \) |
| 13 | \( 1 + 4.47T + 13T^{2} \) |
| 19 | \( 1 - 2.47T + 19T^{2} \) |
| 23 | \( 1 - 3.23T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.23T + 31T^{2} \) |
| 37 | \( 1 - 6.94T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.94T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 5.52T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 4.76T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + 1.70T + 79T^{2} \) |
| 83 | \( 1 - 10.4T + 83T^{2} \) |
| 89 | \( 1 + 16.4T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−12.95499863335649123679834245047, −11.88176029212561145998688564212, −11.40819645886218929442505664158, −10.29935346640717734907909684785, −9.440210581016239559833347742153, −7.54227009460585271658092107954, −6.44153995685194095508342264679, −5.37445701147005362698510239113, −4.62453500205073604857732525057, −1.50989576196998004253019603835,
1.50989576196998004253019603835, 4.62453500205073604857732525057, 5.37445701147005362698510239113, 6.44153995685194095508342264679, 7.54227009460585271658092107954, 9.440210581016239559833347742153, 10.29935346640717734907909684785, 11.40819645886218929442505664158, 11.88176029212561145998688564212, 12.95499863335649123679834245047
