| L(s) = 1 | + 2-s + 4-s + 8-s + 4·11-s − 6·13-s + 16-s + 2·17-s + 4·19-s + 4·22-s − 8·23-s − 6·26-s − 6·29-s − 31-s + 32-s + 2·34-s + 2·37-s + 4·38-s − 10·41-s + 4·43-s + 4·44-s − 8·46-s − 7·49-s − 6·52-s − 10·53-s − 6·58-s + 12·59-s − 2·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s + 1.20·11-s − 1.66·13-s + 1/4·16-s + 0.485·17-s + 0.917·19-s + 0.852·22-s − 1.66·23-s − 1.17·26-s − 1.11·29-s − 0.179·31-s + 0.176·32-s + 0.342·34-s + 0.328·37-s + 0.648·38-s − 1.56·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 49-s − 0.832·52-s − 1.37·53-s − 0.787·58-s + 1.56·59-s − 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 13950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−16.38526124906161, −15.93001880358045, −15.08269132104072, −14.63349762708984, −14.31737625893418, −13.75099636509713, −13.08248369284486, −12.34763640311191, −11.98176985292825, −11.63667978450583, −10.89024157869567, −9.997537792209639, −9.677761800998196, −9.148188602528352, −8.006359536004538, −7.738524454567090, −6.878279426778346, −6.491930263050544, −5.531421723238060, −5.198465352870804, −4.279841125608432, −3.770039839946871, −3.001517573744101, −2.120866049503669, −1.401315602978900, 0,
1.401315602978900, 2.120866049503669, 3.001517573744101, 3.770039839946871, 4.279841125608432, 5.198465352870804, 5.531421723238060, 6.491930263050544, 6.878279426778346, 7.738524454567090, 8.006359536004538, 9.148188602528352, 9.677761800998196, 9.997537792209639, 10.89024157869567, 11.63667978450583, 11.98176985292825, 12.34763640311191, 13.08248369284486, 13.75099636509713, 14.31737625893418, 14.63349762708984, 15.08269132104072, 15.93001880358045, 16.38526124906161
