| L(s) = 1 | − 2-s − 3·3-s + 4-s + 5-s + 3·6-s − 7-s − 8-s + 6·9-s − 10-s + 2·11-s − 3·12-s + 14-s − 3·15-s + 16-s − 3·17-s − 6·18-s − 6·19-s + 20-s + 3·21-s − 2·22-s − 4·23-s + 3·24-s − 4·25-s − 9·27-s − 28-s + 2·29-s + 3·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s − 0.377·7-s − 0.353·8-s + 2·9-s − 0.316·10-s + 0.603·11-s − 0.866·12-s + 0.267·14-s − 0.774·15-s + 1/4·16-s − 0.727·17-s − 1.41·18-s − 1.37·19-s + 0.223·20-s + 0.654·21-s − 0.426·22-s − 0.834·23-s + 0.612·24-s − 4/5·25-s − 1.73·27-s − 0.188·28-s + 0.371·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 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
−11.04319412119313584483866781888, −10.24325458310862263428196114864, −9.539278397314208258267879412937, −8.310061634639968993764403624165, −6.79038227196567694631715258566, −6.40941752007862407532170431759, −5.41501572629242590776107371268, −4.09964629322041262514106195672, −1.82604108038319412812644968804, 0,
1.82604108038319412812644968804, 4.09964629322041262514106195672, 5.41501572629242590776107371268, 6.40941752007862407532170431759, 6.79038227196567694631715258566, 8.310061634639968993764403624165, 9.539278397314208258267879412937, 10.24325458310862263428196114864, 11.04319412119313584483866781888
