| L(s) = 1 | − 3-s − 2·7-s + 9-s − 4·13-s + 6·17-s − 4·19-s + 2·21-s + 6·23-s − 5·25-s − 27-s + 6·29-s + 8·31-s + 10·37-s + 4·39-s − 6·41-s + 8·43-s − 6·47-s − 3·49-s − 6·51-s + 4·57-s + 8·61-s − 2·63-s + 4·67-s − 6·69-s + 6·71-s − 2·73-s + 5·75-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.10·13-s + 1.45·17-s − 0.917·19-s + 0.436·21-s + 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 1.64·37-s + 0.640·39-s − 0.937·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 0.840·51-s + 0.529·57-s + 1.02·61-s − 0.251·63-s + 0.488·67-s − 0.722·69-s + 0.712·71-s − 0.234·73-s + 0.577·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 23232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.333374487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.333374487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.58175686392737, −14.92972186164853, −14.42276780386654, −13.91678864289426, −12.96587652919719, −12.87954131236200, −12.21868801643870, −11.68039413661287, −11.24086698647830, −10.35085561673673, −9.895655538285737, −9.750389109866077, −8.841265258960583, −8.116837321751180, −7.610374457568187, −6.894011743562257, −6.399265769769514, −5.843717633559277, −5.113576960271904, −4.580740292573273, −3.859288921150765, −2.951559910398094, −2.519488677752431, −1.319619414161837, −0.5144853988055250,
0.5144853988055250, 1.319619414161837, 2.519488677752431, 2.951559910398094, 3.859288921150765, 4.580740292573273, 5.113576960271904, 5.843717633559277, 6.399265769769514, 6.894011743562257, 7.610374457568187, 8.116837321751180, 8.841265258960583, 9.750389109866077, 9.895655538285737, 10.35085561673673, 11.24086698647830, 11.68039413661287, 12.21868801643870, 12.87954131236200, 12.96587652919719, 13.91678864289426, 14.42276780386654, 14.92972186164853, 15.58175686392737
