| L(s) = 1 | + 2-s + 4-s − 5-s + 8-s − 10-s − 2·11-s + 16-s + 17-s + 2·19-s − 20-s − 2·22-s + 2·23-s + 25-s + 2·29-s + 32-s + 34-s + 2·37-s + 2·38-s − 40-s + 8·43-s − 2·44-s + 2·46-s + 8·47-s − 7·49-s + 50-s + 4·53-s + 2·55-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.353·8-s − 0.316·10-s − 0.603·11-s + 1/4·16-s + 0.242·17-s + 0.458·19-s − 0.223·20-s − 0.426·22-s + 0.417·23-s + 1/5·25-s + 0.371·29-s + 0.176·32-s + 0.171·34-s + 0.328·37-s + 0.324·38-s − 0.158·40-s + 1.21·43-s − 0.301·44-s + 0.294·46-s + 1.16·47-s − 49-s + 0.141·50-s + 0.549·53-s + 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 258570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−12.99153288316506, −12.63999913888279, −12.10934776534849, −11.85000844841861, −11.20462077324592, −10.83923354963989, −10.45307700028396, −9.906184131060935, −9.288089124764080, −8.941632101717922, −8.168501103032101, −7.870120272389268, −7.379850056571307, −6.946617093706779, −6.360810298825623, −5.797621382566603, −5.428819666499322, −4.753360743560219, −4.503359160685325, −3.737027503383825, −3.399108918519235, −2.638272574719201, −2.422813480376709, −1.430749505369599, −0.8936579164348228, 0,
0.8936579164348228, 1.430749505369599, 2.422813480376709, 2.638272574719201, 3.399108918519235, 3.737027503383825, 4.503359160685325, 4.753360743560219, 5.428819666499322, 5.797621382566603, 6.360810298825623, 6.946617093706779, 7.379850056571307, 7.870120272389268, 8.168501103032101, 8.941632101717922, 9.288089124764080, 9.906184131060935, 10.45307700028396, 10.83923354963989, 11.20462077324592, 11.85000844841861, 12.10934776534849, 12.63999913888279, 12.99153288316506
