L(s) = 1 | + 2-s + 3·3-s + 4-s + 5-s + 3·6-s + 8-s + 6·9-s + 10-s − 2·11-s + 3·12-s + 5·13-s + 3·15-s + 16-s − 17-s + 6·18-s − 2·19-s + 20-s − 2·22-s − 6·23-s + 3·24-s − 4·25-s + 5·26-s + 9·27-s − 6·29-s + 3·30-s − 6·31-s + 32-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.447·5-s + 1.22·6-s + 0.353·8-s + 2·9-s + 0.316·10-s − 0.603·11-s + 0.866·12-s + 1.38·13-s + 0.774·15-s + 1/4·16-s − 0.242·17-s + 1.41·18-s − 0.458·19-s + 0.223·20-s − 0.426·22-s − 1.25·23-s + 0.612·24-s − 4/5·25-s + 0.980·26-s + 1.73·27-s − 1.11·29-s + 0.547·30-s − 1.07·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.013443853\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.013443853\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 5 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 - 15 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.385555119795961536176565306829, −8.627411312705772481690849541929, −7.997263327397438481032071726964, −7.21988974503080386029856198120, −6.19943265191290908083249459089, −5.34056591288281277790160763949, −3.90514322417009055251992921503, −3.67904933564915760697075297181, −2.38397129396486266574988791734, −1.79202010235030914336941155568,
1.79202010235030914336941155568, 2.38397129396486266574988791734, 3.67904933564915760697075297181, 3.90514322417009055251992921503, 5.34056591288281277790160763949, 6.19943265191290908083249459089, 7.21988974503080386029856198120, 7.997263327397438481032071726964, 8.627411312705772481690849541929, 9.385555119795961536176565306829