L(s) = 1 | − 2-s + 4-s − 5-s + 2·7-s − 8-s + 10-s − 4·13-s − 2·14-s + 16-s + 17-s − 4·19-s − 20-s + 25-s + 4·26-s + 2·28-s − 6·29-s − 4·31-s − 32-s − 34-s − 2·35-s + 2·37-s + 4·38-s + 40-s + 2·43-s − 3·49-s − 50-s − 4·52-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s + 0.755·7-s − 0.353·8-s + 0.316·10-s − 1.10·13-s − 0.534·14-s + 1/4·16-s + 0.242·17-s − 0.917·19-s − 0.223·20-s + 1/5·25-s + 0.784·26-s + 0.377·28-s − 1.11·29-s − 0.718·31-s − 0.176·32-s − 0.171·34-s − 0.338·35-s + 0.328·37-s + 0.648·38-s + 0.158·40-s + 0.304·43-s − 3/7·49-s − 0.141·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1530 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 17 | \( 1 - T \) |
good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−9.062768010948203852793619856758, −8.190885549450828256658813783292, −7.60443984278777067688587271846, −6.93439418815201615494528069017, −5.81883976600731049774743123462, −4.88219975867724665721913808894, −3.94620834878742004272258447586, −2.66245115412807667844263553225, −1.62441740500090931646073271114, 0,
1.62441740500090931646073271114, 2.66245115412807667844263553225, 3.94620834878742004272258447586, 4.88219975867724665721913808894, 5.81883976600731049774743123462, 6.93439418815201615494528069017, 7.60443984278777067688587271846, 8.190885549450828256658813783292, 9.062768010948203852793619856758