| L(s) = 1 | − 3-s + 3·5-s + 7-s + 9-s − 11-s + 13-s − 3·15-s − 2·19-s − 21-s − 3·23-s + 4·25-s − 27-s − 3·29-s − 8·31-s + 33-s + 3·35-s + 2·37-s − 39-s − 9·41-s − 5·43-s + 3·45-s − 6·49-s − 6·53-s − 3·55-s + 2·57-s + 3·59-s − 61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.34·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s − 0.774·15-s − 0.458·19-s − 0.218·21-s − 0.625·23-s + 4/5·25-s − 0.192·27-s − 0.557·29-s − 1.43·31-s + 0.174·33-s + 0.507·35-s + 0.328·37-s − 0.160·39-s − 1.40·41-s − 0.762·43-s + 0.447·45-s − 6/7·49-s − 0.824·53-s − 0.404·55-s + 0.264·57-s + 0.390·59-s − 0.128·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−7.53953230655569550336191437205, −6.74408019293904499286334828471, −6.11292823185018654929332216860, −5.53808114020207567026979673012, −4.99523696721203484345897079591, −4.10177495269232651705619041235, −3.10114627748186314117182550441, −1.96735022238452759822188709284, −1.54983453144023684588265796556, 0,
1.54983453144023684588265796556, 1.96735022238452759822188709284, 3.10114627748186314117182550441, 4.10177495269232651705619041235, 4.99523696721203484345897079591, 5.53808114020207567026979673012, 6.11292823185018654929332216860, 6.74408019293904499286334828471, 7.53953230655569550336191437205
