| L(s) = 1 | − 1.11·5-s + 7-s − 1.88·11-s − 13-s − 5.87·17-s + 7.09·19-s + 3.98·23-s − 3.76·25-s + 0.987·29-s − 0.666·31-s − 1.11·35-s − 1.33·37-s − 1.88·41-s − 10.8·43-s − 11.0·47-s + 49-s − 12.2·53-s + 2.09·55-s + 4.76·59-s + 3.76·61-s + 1.11·65-s + 4.09·67-s − 10.7·71-s − 3.09·73-s − 1.88·77-s + 6.43·79-s − 11.8·83-s + ⋯ |
| L(s) = 1 | − 0.496·5-s + 0.377·7-s − 0.569·11-s − 0.277·13-s − 1.42·17-s + 1.62·19-s + 0.831·23-s − 0.753·25-s + 0.183·29-s − 0.119·31-s − 0.187·35-s − 0.219·37-s − 0.294·41-s − 1.65·43-s − 1.61·47-s + 0.142·49-s − 1.67·53-s + 0.283·55-s + 0.620·59-s + 0.482·61-s + 0.137·65-s + 0.500·67-s − 1.27·71-s − 0.362·73-s − 0.215·77-s + 0.723·79-s − 1.30·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2268 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 5 | \( 1 + 1.11T + 5T^{2} \) |
| 11 | \( 1 + 1.88T + 11T^{2} \) |
| 13 | \( 1 + T + 13T^{2} \) |
| 17 | \( 1 + 5.87T + 17T^{2} \) |
| 19 | \( 1 - 7.09T + 19T^{2} \) |
| 23 | \( 1 - 3.98T + 23T^{2} \) |
| 29 | \( 1 - 0.987T + 29T^{2} \) |
| 31 | \( 1 + 0.666T + 31T^{2} \) |
| 37 | \( 1 + 1.33T + 37T^{2} \) |
| 41 | \( 1 + 1.88T + 41T^{2} \) |
| 43 | \( 1 + 10.8T + 43T^{2} \) |
| 47 | \( 1 + 11.0T + 47T^{2} \) |
| 53 | \( 1 + 12.2T + 53T^{2} \) |
| 59 | \( 1 - 4.76T + 59T^{2} \) |
| 61 | \( 1 - 3.76T + 61T^{2} \) |
| 67 | \( 1 - 4.09T + 67T^{2} \) |
| 71 | \( 1 + 10.7T + 71T^{2} \) |
| 73 | \( 1 + 3.09T + 73T^{2} \) |
| 79 | \( 1 - 6.43T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 14.3T + 89T^{2} \) |
| 97 | \( 1 - 0.765T + 97T^{2} \) |
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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
−8.530681406361361359595015378755, −7.88169095658946503608805081964, −7.17396742873843156294257152477, −6.41477882150252775699833756528, −5.21103282650892653691358034303, −4.79123905303136137757388077764, −3.66197262649540127592504959228, −2.78089641832987926332701048230, −1.58110650063281901945265179051, 0,
1.58110650063281901945265179051, 2.78089641832987926332701048230, 3.66197262649540127592504959228, 4.79123905303136137757388077764, 5.21103282650892653691358034303, 6.41477882150252775699833756528, 7.17396742873843156294257152477, 7.88169095658946503608805081964, 8.530681406361361359595015378755
