| L(s) = 1 | + 2·2-s + 4·4-s − 8·7-s + 8·8-s − 18·11-s − 8·13-s − 16·14-s + 16·16-s + 15·17-s + 23·19-s − 36·22-s + 63·23-s − 16·26-s − 32·28-s − 156·29-s − 85·31-s + 32·32-s + 30·34-s − 74·37-s + 46·38-s − 246·41-s + 190·43-s − 72·44-s + 126·46-s + 288·47-s − 279·49-s − 32·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.431·7-s + 0.353·8-s − 0.493·11-s − 0.170·13-s − 0.305·14-s + 1/4·16-s + 0.214·17-s + 0.277·19-s − 0.348·22-s + 0.571·23-s − 0.120·26-s − 0.215·28-s − 0.998·29-s − 0.492·31-s + 0.176·32-s + 0.151·34-s − 0.328·37-s + 0.196·38-s − 0.937·41-s + 0.673·43-s − 0.246·44-s + 0.403·46-s + 0.893·47-s − 0.813·49-s − 0.0853·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 18 T + p^{3} T^{2} \) |
| 13 | \( 1 + 8 T + p^{3} T^{2} \) |
| 17 | \( 1 - 15 T + p^{3} T^{2} \) |
| 19 | \( 1 - 23 T + p^{3} T^{2} \) |
| 23 | \( 1 - 63 T + p^{3} T^{2} \) |
| 29 | \( 1 + 156 T + p^{3} T^{2} \) |
| 31 | \( 1 + 85 T + p^{3} T^{2} \) |
| 37 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 41 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 43 | \( 1 - 190 T + p^{3} T^{2} \) |
| 47 | \( 1 - 288 T + p^{3} T^{2} \) |
| 53 | \( 1 + 177 T + p^{3} T^{2} \) |
| 59 | \( 1 + 792 T + p^{3} T^{2} \) |
| 61 | \( 1 + 907 T + p^{3} T^{2} \) |
| 67 | \( 1 - 322 T + p^{3} T^{2} \) |
| 71 | \( 1 - 270 T + p^{3} T^{2} \) |
| 73 | \( 1 + 254 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1123 T + p^{3} T^{2} \) |
| 83 | \( 1 + 771 T + p^{3} T^{2} \) |
| 89 | \( 1 - 198 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1192 T + p^{3} T^{2} \) |
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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
−8.905045841125205127692246680779, −7.77496375257808431806314033963, −7.18192218334510411020114943460, −6.20957084608211217463756250374, −5.44669624962899704844979852643, −4.62085253026115518416364271784, −3.54442754384111764554967686845, −2.78012610208909385500061434182, −1.56152716555965960974207567310, 0,
1.56152716555965960974207567310, 2.78012610208909385500061434182, 3.54442754384111764554967686845, 4.62085253026115518416364271784, 5.44669624962899704844979852643, 6.20957084608211217463756250374, 7.18192218334510411020114943460, 7.77496375257808431806314033963, 8.905045841125205127692246680779
