| L(s) = 1 | + 1.56·3-s + 5-s − 2.56·7-s − 0.561·9-s − 2·11-s − 3.56·13-s + 1.56·15-s + 2.56·17-s − 6·19-s − 4·21-s − 23-s + 25-s − 5.56·27-s + 6.12·29-s − 7.24·31-s − 3.12·33-s − 2.56·35-s − 4.56·37-s − 5.56·39-s + 4.12·41-s − 0.561·45-s − 4.68·47-s − 0.438·49-s + 4·51-s − 4.56·53-s − 2·55-s − 9.36·57-s + ⋯ |
| L(s) = 1 | + 0.901·3-s + 0.447·5-s − 0.968·7-s − 0.187·9-s − 0.603·11-s − 0.987·13-s + 0.403·15-s + 0.621·17-s − 1.37·19-s − 0.872·21-s − 0.208·23-s + 0.200·25-s − 1.07·27-s + 1.13·29-s − 1.30·31-s − 0.543·33-s − 0.432·35-s − 0.749·37-s − 0.890·39-s + 0.643·41-s − 0.0837·45-s − 0.683·47-s − 0.0626·49-s + 0.560·51-s − 0.626·53-s − 0.269·55-s − 1.24·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 1.56T + 3T^{2} \) |
| 7 | \( 1 + 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 3.56T + 13T^{2} \) |
| 17 | \( 1 - 2.56T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 29 | \( 1 - 6.12T + 29T^{2} \) |
| 31 | \( 1 + 7.24T + 31T^{2} \) |
| 37 | \( 1 + 4.56T + 37T^{2} \) |
| 41 | \( 1 - 4.12T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 4.68T + 47T^{2} \) |
| 53 | \( 1 + 4.56T + 53T^{2} \) |
| 59 | \( 1 - 3.68T + 59T^{2} \) |
| 61 | \( 1 + 7.12T + 61T^{2} \) |
| 67 | \( 1 - 8.56T + 67T^{2} \) |
| 71 | \( 1 + 10.1T + 71T^{2} \) |
| 73 | \( 1 + 4.43T + 73T^{2} \) |
| 79 | \( 1 + 4.87T + 79T^{2} \) |
| 83 | \( 1 - 13.9T + 83T^{2} \) |
| 89 | \( 1 - 14.2T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.901544880561462316381874466877, −8.159414376033163239288205084303, −7.38307302199769513458956062097, −6.48470323780968505175660361376, −5.70038962179969019640991802601, −4.72781233605554511483273136684, −3.54242697742674730233247379829, −2.79652366704714863587821808439, −2.01861015632329204615327054359, 0,
2.01861015632329204615327054359, 2.79652366704714863587821808439, 3.54242697742674730233247379829, 4.72781233605554511483273136684, 5.70038962179969019640991802601, 6.48470323780968505175660361376, 7.38307302199769513458956062097, 8.159414376033163239288205084303, 8.901544880561462316381874466877
