| L(s) = 1 | − 2-s + 4-s + 2·5-s − 7-s − 8-s − 2·10-s − 2·11-s − 6·13-s + 14-s + 16-s + 4·17-s + 19-s + 2·20-s + 2·22-s + 4·23-s − 25-s + 6·26-s − 28-s + 2·29-s − 6·31-s − 32-s − 4·34-s − 2·35-s − 4·37-s − 38-s − 2·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.603·11-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.970·17-s + 0.229·19-s + 0.447·20-s + 0.426·22-s + 0.834·23-s − 1/5·25-s + 1.17·26-s − 0.188·28-s + 0.371·29-s − 1.07·31-s − 0.176·32-s − 0.685·34-s − 0.338·35-s − 0.657·37-s − 0.162·38-s − 0.316·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−8.788916774413033249547378820074, −7.63641766051590233056548750771, −7.30291670832811286525122598537, −6.31562908342391033302623846390, −5.47483278431711319143114344793, −4.86907198994118583089745626209, −3.32051948332236639619472874169, −2.54607376327911572141600580767, −1.56379247334588687287752665516, 0,
1.56379247334588687287752665516, 2.54607376327911572141600580767, 3.32051948332236639619472874169, 4.86907198994118583089745626209, 5.47483278431711319143114344793, 6.31562908342391033302623846390, 7.30291670832811286525122598537, 7.63641766051590233056548750771, 8.788916774413033249547378820074
