| L(s) = 1 | + 2·2-s − 3·3-s − 4·4-s + 5·5-s − 6·6-s − 2·7-s − 24·8-s − 18·9-s + 10·10-s − 16·11-s + 12·12-s − 47·13-s − 4·14-s − 15·15-s − 16·16-s − 24·17-s − 36·18-s − 56·19-s − 20·20-s + 6·21-s − 32·22-s − 23·23-s + 72·24-s + 25·25-s − 94·26-s + 135·27-s + 8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.107·7-s − 1.06·8-s − 2/3·9-s + 0.316·10-s − 0.438·11-s + 0.288·12-s − 1.00·13-s − 0.0763·14-s − 0.258·15-s − 1/4·16-s − 0.342·17-s − 0.471·18-s − 0.676·19-s − 0.223·20-s + 0.0623·21-s − 0.310·22-s − 0.208·23-s + 0.612·24-s + 1/5·25-s − 0.709·26-s + 0.962·27-s + 0.0539·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - p T \) |
| 23 | \( 1 + p T \) |
| good | 2 | \( 1 - p T + p^{3} T^{2} \) |
| 3 | \( 1 + p T + p^{3} T^{2} \) |
| 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 + 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 47 T + p^{3} T^{2} \) |
| 17 | \( 1 + 24 T + p^{3} T^{2} \) |
| 19 | \( 1 + 56 T + p^{3} T^{2} \) |
| 29 | \( 1 - 85 T + p^{3} T^{2} \) |
| 31 | \( 1 - 67 T + p^{3} T^{2} \) |
| 37 | \( 1 - 104 T + p^{3} T^{2} \) |
| 41 | \( 1 + 53 T + p^{3} T^{2} \) |
| 43 | \( 1 + 234 T + p^{3} T^{2} \) |
| 47 | \( 1 - 285 T + p^{3} T^{2} \) |
| 53 | \( 1 - 2 T + p^{3} T^{2} \) |
| 59 | \( 1 - 80 T + p^{3} T^{2} \) |
| 61 | \( 1 + 764 T + p^{3} T^{2} \) |
| 67 | \( 1 - 236 T + p^{3} T^{2} \) |
| 71 | \( 1 + 289 T + p^{3} T^{2} \) |
| 73 | \( 1 + 225 T + p^{3} T^{2} \) |
| 79 | \( 1 - 24 T + p^{3} T^{2} \) |
| 83 | \( 1 - 684 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1370 T + p^{3} T^{2} \) |
| 97 | \( 1 + 110 T + p^{3} T^{2} \) |
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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
−12.62473830138726021822781319226, −11.81738890813518215519086870923, −10.54115165425474789150963672913, −9.438991490602391872746786499377, −8.278925163476720277372220008673, −6.52676103716535718236934152456, −5.50555403756233399211378041138, −4.56036285838337283133747946354, −2.77522407183088203267266037926, 0,
2.77522407183088203267266037926, 4.56036285838337283133747946354, 5.50555403756233399211378041138, 6.52676103716535718236934152456, 8.278925163476720277372220008673, 9.438991490602391872746786499377, 10.54115165425474789150963672913, 11.81738890813518215519086870923, 12.62473830138726021822781319226
