| L(s) = 1 | + 2-s + 4-s − 5-s + 4·7-s + 8-s − 10-s + 6·11-s − 6·13-s + 4·14-s + 16-s + 3·17-s − 6·19-s − 20-s + 6·22-s + 7·23-s − 4·25-s − 6·26-s + 4·28-s + 6·31-s + 32-s + 3·34-s − 4·35-s − 3·37-s − 6·38-s − 40-s + 7·43-s + 6·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.51·7-s + 0.353·8-s − 0.316·10-s + 1.80·11-s − 1.66·13-s + 1.06·14-s + 1/4·16-s + 0.727·17-s − 1.37·19-s − 0.223·20-s + 1.27·22-s + 1.45·23-s − 4/5·25-s − 1.17·26-s + 0.755·28-s + 1.07·31-s + 0.176·32-s + 0.514·34-s − 0.676·35-s − 0.493·37-s − 0.973·38-s − 0.158·40-s + 1.06·43-s + 0.904·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1098 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.856351741\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.856351741\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 61 | \( 1 + T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.917461859219297653369521903530, −8.954461847508739444883118593331, −8.069357192077553612030161472406, −7.30055553795352770723922990587, −6.53709602046399974447217609868, −5.34558974721025801567566921767, −4.54876985837096746161070305611, −3.95885180382732927496948309529, −2.52097089588598704365976417676, −1.35091118273881221529817713534,
1.35091118273881221529817713534, 2.52097089588598704365976417676, 3.95885180382732927496948309529, 4.54876985837096746161070305611, 5.34558974721025801567566921767, 6.53709602046399974447217609868, 7.30055553795352770723922990587, 8.069357192077553612030161472406, 8.954461847508739444883118593331, 9.917461859219297653369521903530
