| L(s) = 1 | + 15·3-s + 19·5-s + 10·7-s − 18·9-s + 121·11-s + 1.14e3·13-s + 285·15-s + 686·17-s + 384·19-s + 150·21-s + 3.70e3·23-s − 2.76e3·25-s − 3.91e3·27-s + 5.42e3·29-s − 6.44e3·31-s + 1.81e3·33-s + 190·35-s − 1.20e4·37-s + 1.72e4·39-s − 1.52e3·41-s + 4.02e3·43-s − 342·45-s + 7.16e3·47-s − 1.67e4·49-s + 1.02e4·51-s + 2.98e4·53-s + 2.29e3·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s + 0.339·5-s + 0.0771·7-s − 0.0740·9-s + 0.301·11-s + 1.88·13-s + 0.327·15-s + 0.575·17-s + 0.244·19-s + 0.0742·21-s + 1.46·23-s − 0.884·25-s − 1.03·27-s + 1.19·29-s − 1.20·31-s + 0.290·33-s + 0.0262·35-s − 1.44·37-s + 1.81·39-s − 0.141·41-s + 0.332·43-s − 0.0251·45-s + 0.473·47-s − 0.994·49-s + 0.553·51-s + 1.46·53-s + 0.102·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 704 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.029570992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.029570992\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - p^{2} T \) |
| good | 3 | \( 1 - 5 p T + p^{5} T^{2} \) |
| 5 | \( 1 - 19 T + p^{5} T^{2} \) |
| 7 | \( 1 - 10 T + p^{5} T^{2} \) |
| 13 | \( 1 - 1148 T + p^{5} T^{2} \) |
| 17 | \( 1 - 686 T + p^{5} T^{2} \) |
| 19 | \( 1 - 384 T + p^{5} T^{2} \) |
| 23 | \( 1 - 3709 T + p^{5} T^{2} \) |
| 29 | \( 1 - 5424 T + p^{5} T^{2} \) |
| 31 | \( 1 + 6443 T + p^{5} T^{2} \) |
| 37 | \( 1 + 12063 T + p^{5} T^{2} \) |
| 41 | \( 1 + 1528 T + p^{5} T^{2} \) |
| 43 | \( 1 - 4026 T + p^{5} T^{2} \) |
| 47 | \( 1 - 7168 T + p^{5} T^{2} \) |
| 53 | \( 1 - 29862 T + p^{5} T^{2} \) |
| 59 | \( 1 - 6461 T + p^{5} T^{2} \) |
| 61 | \( 1 - 16980 T + p^{5} T^{2} \) |
| 67 | \( 1 + 29999 T + p^{5} T^{2} \) |
| 71 | \( 1 - 31023 T + p^{5} T^{2} \) |
| 73 | \( 1 - 1924 T + p^{5} T^{2} \) |
| 79 | \( 1 - 65138 T + p^{5} T^{2} \) |
| 83 | \( 1 - 102714 T + p^{5} T^{2} \) |
| 89 | \( 1 - 17415 T + p^{5} T^{2} \) |
| 97 | \( 1 - 66905 T + p^{5} T^{2} \) |
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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
−9.416311509149669661655178899332, −8.788426148944007130220966248199, −8.201412538180997764954462953428, −7.14123536966973486765135282406, −6.12036958068403336438219252989, −5.25340890996566383191006142377, −3.78187894308196582478661836986, −3.21924843064313731944118816260, −1.96326054024768087122368911313, −0.947438729102886250956308405437,
0.947438729102886250956308405437, 1.96326054024768087122368911313, 3.21924843064313731944118816260, 3.78187894308196582478661836986, 5.25340890996566383191006142377, 6.12036958068403336438219252989, 7.14123536966973486765135282406, 8.201412538180997764954462953428, 8.788426148944007130220966248199, 9.416311509149669661655178899332
