L(s) = 1 | − 2-s − 3-s − 4-s − 5-s + 6-s − 4·7-s + 3·8-s − 2·9-s + 10-s − 2·11-s + 12-s + 2·13-s + 4·14-s + 15-s − 16-s + 3·17-s + 2·18-s − 5·19-s + 20-s + 4·21-s + 2·22-s + 7·23-s − 3·24-s − 4·25-s − 2·26-s + 5·27-s + 4·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.447·5-s + 0.408·6-s − 1.51·7-s + 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 0.554·13-s + 1.06·14-s + 0.258·15-s − 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.14·19-s + 0.223·20-s + 0.872·21-s + 0.426·22-s + 1.45·23-s − 0.612·24-s − 4/5·25-s − 0.392·26-s + 0.962·27-s + 0.755·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 89 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 89 | \( 1 + T \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.31042167272302452953280111333, −12.68407335930173320446176180679, −11.21974022600404577936134169212, −10.27280155411412996073709887551, −9.201725180013691531910215134196, −8.158202871876967356622403997141, −6.70828498198506304645222570612, −5.33655779421800086996780386264, −3.50979110381215119742449031067, 0,
3.50979110381215119742449031067, 5.33655779421800086996780386264, 6.70828498198506304645222570612, 8.158202871876967356622403997141, 9.201725180013691531910215134196, 10.27280155411412996073709887551, 11.21974022600404577936134169212, 12.68407335930173320446176180679, 13.31042167272302452953280111333