| L(s) = 1 | + 26·3-s + 16·5-s − 49·7-s + 433·9-s + 8·11-s + 684·13-s + 416·15-s − 2.21e3·17-s − 2.69e3·19-s − 1.27e3·21-s + 3.34e3·23-s − 2.86e3·25-s + 4.94e3·27-s − 3.25e3·29-s + 4.78e3·31-s + 208·33-s − 784·35-s − 1.14e4·37-s + 1.77e4·39-s + 1.33e4·41-s − 928·43-s + 6.92e3·45-s + 1.21e3·47-s + 2.40e3·49-s − 5.76e4·51-s + 1.31e4·53-s + 128·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s + 0.286·5-s − 0.377·7-s + 1.78·9-s + 0.0199·11-s + 1.12·13-s + 0.477·15-s − 1.86·17-s − 1.71·19-s − 0.630·21-s + 1.31·23-s − 0.918·25-s + 1.30·27-s − 0.718·29-s + 0.894·31-s + 0.0332·33-s − 0.108·35-s − 1.37·37-s + 1.87·39-s + 1.24·41-s − 0.0765·43-s + 0.510·45-s + 0.0800·47-s + 1/7·49-s − 3.10·51-s + 0.641·53-s + 0.00570·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.375874587\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.375874587\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{2} T \) |
| good | 3 | \( 1 - 26 T + p^{5} T^{2} \) |
| 5 | \( 1 - 16 T + p^{5} T^{2} \) |
| 11 | \( 1 - 8 T + p^{5} T^{2} \) |
| 13 | \( 1 - 684 T + p^{5} T^{2} \) |
| 17 | \( 1 + 2218 T + p^{5} T^{2} \) |
| 19 | \( 1 + 142 p T + p^{5} T^{2} \) |
| 23 | \( 1 - 3344 T + p^{5} T^{2} \) |
| 29 | \( 1 + 3254 T + p^{5} T^{2} \) |
| 31 | \( 1 - 4788 T + p^{5} T^{2} \) |
| 37 | \( 1 + 310 p T + p^{5} T^{2} \) |
| 41 | \( 1 - 13350 T + p^{5} T^{2} \) |
| 43 | \( 1 + 928 T + p^{5} T^{2} \) |
| 47 | \( 1 - 1212 T + p^{5} T^{2} \) |
| 53 | \( 1 - 13110 T + p^{5} T^{2} \) |
| 59 | \( 1 - 34702 T + p^{5} T^{2} \) |
| 61 | \( 1 + 1032 T + p^{5} T^{2} \) |
| 67 | \( 1 - 10108 T + p^{5} T^{2} \) |
| 71 | \( 1 - 62720 T + p^{5} T^{2} \) |
| 73 | \( 1 + 18926 T + p^{5} T^{2} \) |
| 79 | \( 1 - 11400 T + p^{5} T^{2} \) |
| 83 | \( 1 - 88958 T + p^{5} T^{2} \) |
| 89 | \( 1 - 19722 T + p^{5} T^{2} \) |
| 97 | \( 1 - 17062 T + p^{5} T^{2} \) |
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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
−15.75091300029627238404180851564, −14.96456783650792418454325624854, −13.59293478606349901112780335961, −13.04267118646930709050283206856, −10.79950993983158190076935110077, −9.206021531720503906539246669874, −8.439833948633120338922246953551, −6.67692715171034320407584043685, −3.95791997940940384003663737110, −2.24537971834612380657569247966,
2.24537971834612380657569247966, 3.95791997940940384003663737110, 6.67692715171034320407584043685, 8.439833948633120338922246953551, 9.206021531720503906539246669874, 10.79950993983158190076935110077, 13.04267118646930709050283206856, 13.59293478606349901112780335961, 14.96456783650792418454325624854, 15.75091300029627238404180851564
