L(s) = 1 | + 2-s + 4-s + 2·7-s + 8-s − 2·11-s + 4·13-s + 2·14-s + 16-s − 6·17-s + 19-s − 2·22-s − 8·23-s + 4·26-s + 2·28-s − 6·29-s − 8·31-s + 32-s − 6·34-s + 8·37-s + 38-s − 12·41-s − 2·44-s − 8·46-s − 3·49-s + 4·52-s − 10·53-s + 2·56-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.755·7-s + 0.353·8-s − 0.603·11-s + 1.10·13-s + 0.534·14-s + 1/4·16-s − 1.45·17-s + 0.229·19-s − 0.426·22-s − 1.66·23-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 1.43·31-s + 0.176·32-s − 1.02·34-s + 1.31·37-s + 0.162·38-s − 1.87·41-s − 0.301·44-s − 1.17·46-s − 3/7·49-s + 0.554·52-s − 1.37·53-s + 0.267·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8550 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.55196063058328307116631211251, −6.50720297645715505233428197854, −6.06836678665006477970710088974, −5.28754380955079178244411795767, −4.62939916428607101352862148388, −3.93555935553654627582589151672, −3.24941784356345959590610351224, −2.09698326370647580425927672294, −1.64728875536469951271641601528, 0,
1.64728875536469951271641601528, 2.09698326370647580425927672294, 3.24941784356345959590610351224, 3.93555935553654627582589151672, 4.62939916428607101352862148388, 5.28754380955079178244411795767, 6.06836678665006477970710088974, 6.50720297645715505233428197854, 7.55196063058328307116631211251