| L(s) = 1 | + 2·3-s + 5-s + 9-s − 11-s − 2·13-s + 2·15-s + 2·17-s − 19-s − 3·23-s − 4·25-s − 4·27-s + 4·29-s − 4·31-s − 2·33-s − 10·37-s − 4·39-s − 10·41-s + 43-s + 45-s − 47-s + 4·51-s − 4·53-s − 55-s − 2·57-s − 8·59-s − 3·61-s − 2·65-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.554·13-s + 0.516·15-s + 0.485·17-s − 0.229·19-s − 0.625·23-s − 4/5·25-s − 0.769·27-s + 0.742·29-s − 0.718·31-s − 0.348·33-s − 1.64·37-s − 0.640·39-s − 1.56·41-s + 0.152·43-s + 0.149·45-s − 0.145·47-s + 0.560·51-s − 0.549·53-s − 0.134·55-s − 0.264·57-s − 1.04·59-s − 0.384·61-s − 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + 3 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 3 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 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
−7.77264477811774933236656795852, −6.98629693011491764887378961273, −6.17142998380942075660967844864, −5.41323664970275016132995719855, −4.70335327914616480903288157807, −3.66332064651356135687165805731, −3.15640817634203331784134249058, −2.22570173505304067504246021394, −1.67841841078487855070680846887, 0,
1.67841841078487855070680846887, 2.22570173505304067504246021394, 3.15640817634203331784134249058, 3.66332064651356135687165805731, 4.70335327914616480903288157807, 5.41323664970275016132995719855, 6.17142998380942075660967844864, 6.98629693011491764887378961273, 7.77264477811774933236656795852
