L(s) = 1 | + 2-s + 4-s + 5-s − 7-s + 8-s + 10-s + 2·11-s − 2·13-s − 14-s + 16-s + 6·19-s + 20-s + 2·22-s + 25-s − 2·26-s − 28-s + 8·29-s − 10·31-s + 32-s − 35-s + 10·37-s + 6·38-s + 40-s + 10·41-s − 2·43-s + 2·44-s + 2·47-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.603·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.37·19-s + 0.223·20-s + 0.426·22-s + 1/5·25-s − 0.392·26-s − 0.188·28-s + 1.48·29-s − 1.79·31-s + 0.176·32-s − 0.169·35-s + 1.64·37-s + 0.973·38-s + 0.158·40-s + 1.56·41-s − 0.304·43-s + 0.301·44-s + 0.291·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 333270 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 \) |
good | 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 18 T + p T^{2} \) |
| 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
−12.80114451939054, −12.48410176584913, −11.96875439526923, −11.56075140459437, −11.01075881022064, −10.74077568706228, −9.976624704173598, −9.553296630206024, −9.439712101893470, −8.771289308475785, −8.108098677746914, −7.633545441748792, −7.148460267694937, −6.778919436602276, −6.100984245490925, −5.907857181607745, −5.184236070414809, −4.943129684900749, −4.136559462771581, −3.885491316467157, −3.114049269633255, −2.706413637447777, −2.264677367251196, −1.333040288848000, −1.048028680290590, 0,
1.048028680290590, 1.333040288848000, 2.264677367251196, 2.706413637447777, 3.114049269633255, 3.885491316467157, 4.136559462771581, 4.943129684900749, 5.184236070414809, 5.907857181607745, 6.100984245490925, 6.778919436602276, 7.148460267694937, 7.633545441748792, 8.108098677746914, 8.771289308475785, 9.439712101893470, 9.553296630206024, 9.976624704173598, 10.74077568706228, 11.01075881022064, 11.56075140459437, 11.96875439526923, 12.48410176584913, 12.80114451939054