| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s + 11-s − 2·12-s − 2·13-s − 4·16-s − 4·17-s + 2·18-s + 3·19-s + 2·22-s + 23-s − 5·25-s − 4·26-s − 27-s − 6·29-s + 2·31-s − 8·32-s − 33-s − 8·34-s + 2·36-s − 2·37-s + 6·38-s + 2·39-s − 41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.554·13-s − 16-s − 0.970·17-s + 0.471·18-s + 0.688·19-s + 0.426·22-s + 0.208·23-s − 25-s − 0.784·26-s − 0.192·27-s − 1.11·29-s + 0.359·31-s − 1.41·32-s − 0.174·33-s − 1.37·34-s + 1/3·36-s − 0.328·37-s + 0.973·38-s + 0.320·39-s − 0.156·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.046138257812354808822590314892, −7.10690556999089671090045248972, −6.57440145460962863782823879542, −5.73597596461189287969461125177, −5.19777631073343164429785106963, −4.41458363033730067860891263482, −3.77058893117459408976638851728, −2.79765766313598470514129515131, −1.75642569016547025525615724755, 0,
1.75642569016547025525615724755, 2.79765766313598470514129515131, 3.77058893117459408976638851728, 4.41458363033730067860891263482, 5.19777631073343164429785106963, 5.73597596461189287969461125177, 6.57440145460962863782823879542, 7.10690556999089671090045248972, 8.046138257812354808822590314892
