| L(s) = 1 | − 2-s + 4-s − 2·7-s − 8-s − 2·13-s + 2·14-s + 16-s + 19-s + 2·26-s − 2·28-s + 6·29-s + 2·31-s − 32-s − 2·37-s − 38-s − 8·43-s − 3·49-s − 2·52-s + 6·53-s + 2·56-s − 6·58-s + 6·59-s + 2·61-s − 2·62-s + 64-s + 4·67-s − 14·73-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.755·7-s − 0.353·8-s − 0.554·13-s + 0.534·14-s + 1/4·16-s + 0.229·19-s + 0.392·26-s − 0.377·28-s + 1.11·29-s + 0.359·31-s − 0.176·32-s − 0.328·37-s − 0.162·38-s − 1.21·43-s − 3/7·49-s − 0.277·52-s + 0.824·53-s + 0.267·56-s − 0.787·58-s + 0.781·59-s + 0.256·61-s − 0.254·62-s + 1/8·64-s + 0.488·67-s − 1.63·73-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.45882364780347198919984849622, −6.76523557910195826838134672251, −6.33406074921404190041150736843, −5.42549620843772083333498645624, −4.70242741676313895376986087177, −3.68512195580055265805009096686, −2.95595634536266103510509572323, −2.20031013413264128624288801619, −1.08029552142377791652185581101, 0,
1.08029552142377791652185581101, 2.20031013413264128624288801619, 2.95595634536266103510509572323, 3.68512195580055265805009096686, 4.70242741676313895376986087177, 5.42549620843772083333498645624, 6.33406074921404190041150736843, 6.76523557910195826838134672251, 7.45882364780347198919984849622
