| L(s) = 1 | − 3-s − 2·5-s + 9-s − 11-s + 2·15-s − 6·17-s + 19-s + 23-s − 25-s − 27-s + 4·29-s + 10·31-s + 33-s + 4·37-s + 3·41-s − 12·43-s − 2·45-s + 3·47-s + 6·51-s + 3·53-s + 2·55-s − 57-s + 15·59-s + 3·61-s − 4·67-s − 69-s − 6·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s + 0.516·15-s − 1.45·17-s + 0.229·19-s + 0.208·23-s − 1/5·25-s − 0.192·27-s + 0.742·29-s + 1.79·31-s + 0.174·33-s + 0.657·37-s + 0.468·41-s − 1.82·43-s − 0.298·45-s + 0.437·47-s + 0.840·51-s + 0.412·53-s + 0.269·55-s − 0.132·57-s + 1.95·59-s + 0.384·61-s − 0.488·67-s − 0.120·69-s − 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 10 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 15 T + p T^{2} \) |
| 61 | \( 1 - 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.19002196504679, −12.75142374307582, −12.10185645291850, −11.81365655511354, −11.34891355634758, −11.12149967782741, −10.34387771566899, −10.13795475535970, −9.539919534237457, −8.907061636372781, −8.359538237188636, −8.123791869603033, −7.483281823256242, −6.953466269306931, −6.527818707242059, −6.155446628563321, −5.363264365181217, −4.929750313897650, −4.419426653586319, −4.017646067891043, −3.420250774743131, −2.625978691569099, −2.283026556035684, −1.297427752918586, −0.6728720382980502, 0,
0.6728720382980502, 1.297427752918586, 2.283026556035684, 2.625978691569099, 3.420250774743131, 4.017646067891043, 4.419426653586319, 4.929750313897650, 5.363264365181217, 6.155446628563321, 6.527818707242059, 6.953466269306931, 7.483281823256242, 8.123791869603033, 8.359538237188636, 8.907061636372781, 9.539919534237457, 10.13795475535970, 10.34387771566899, 11.12149967782741, 11.34891355634758, 11.81365655511354, 12.10185645291850, 12.75142374307582, 13.19002196504679
