| L(s) = 1 | + 3-s − 2.49·5-s − 0.917·7-s + 9-s + 3.69·11-s − 5.81·13-s − 2.49·15-s − 0.867·17-s + 6.52·19-s − 0.917·21-s − 4·23-s + 1.23·25-s + 27-s + 7.72·29-s + 2.14·31-s + 3.69·33-s + 2.29·35-s + 2.47·37-s − 5.81·39-s − 9.58·41-s − 9.58·43-s − 2.49·45-s + 1.65·47-s − 6.15·49-s − 0.867·51-s − 3.39·53-s − 9.22·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.11·5-s − 0.346·7-s + 0.333·9-s + 1.11·11-s − 1.61·13-s − 0.644·15-s − 0.210·17-s + 1.49·19-s − 0.200·21-s − 0.834·23-s + 0.246·25-s + 0.192·27-s + 1.43·29-s + 0.385·31-s + 0.643·33-s + 0.387·35-s + 0.406·37-s − 0.930·39-s − 1.49·41-s − 1.46·43-s − 0.372·45-s + 0.241·47-s − 0.879·49-s − 0.121·51-s − 0.466·53-s − 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| good | 5 | \( 1 + 2.49T + 5T^{2} \) |
| 7 | \( 1 + 0.917T + 7T^{2} \) |
| 11 | \( 1 - 3.69T + 11T^{2} \) |
| 13 | \( 1 + 5.81T + 13T^{2} \) |
| 17 | \( 1 + 0.867T + 17T^{2} \) |
| 19 | \( 1 - 6.52T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 7.72T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 2.47T + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 + 9.58T + 43T^{2} \) |
| 47 | \( 1 - 1.65T + 47T^{2} \) |
| 53 | \( 1 + 3.39T + 53T^{2} \) |
| 59 | \( 1 + 12.7T + 59T^{2} \) |
| 61 | \( 1 + 0.0231T + 61T^{2} \) |
| 67 | \( 1 - 5.32T + 67T^{2} \) |
| 71 | \( 1 + 11.8T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 + 8.40T + 79T^{2} \) |
| 83 | \( 1 - 1.96T + 83T^{2} \) |
| 89 | \( 1 - 2.79T + 89T^{2} \) |
| 97 | \( 1 + 2.26T + 97T^{2} \) |
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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
−8.244614797759709973993425014948, −7.61586474138751451266926906175, −7.01662586988163288542684060591, −6.27176399599004218563323955119, −4.96305351731895960539401987293, −4.38015112702172828475138581270, −3.44803261083331813117446767500, −2.84168246940134948622485681486, −1.49424277809741349085726871212, 0,
1.49424277809741349085726871212, 2.84168246940134948622485681486, 3.44803261083331813117446767500, 4.38015112702172828475138581270, 4.96305351731895960539401987293, 6.27176399599004218563323955119, 7.01662586988163288542684060591, 7.61586474138751451266926906175, 8.244614797759709973993425014948
