| L(s) = 1 | − 2-s − 3-s − 4-s + 5-s + 6-s − 7-s + 3·8-s + 9-s − 10-s + 11-s + 12-s − 2·13-s + 14-s − 15-s − 16-s − 6·17-s − 18-s − 4·19-s − 20-s + 21-s − 22-s + 8·23-s − 3·24-s + 25-s + 2·26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s + 0.408·6-s − 0.377·7-s + 1.06·8-s + 1/3·9-s − 0.316·10-s + 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.218·21-s − 0.213·22-s + 1.66·23-s − 0.612·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−19.88527390088876, −19.40426436288095, −18.80154326137500, −18.01599762312436, −17.42174941409241, −17.05626771105392, −16.41093970400270, −15.43500679266512, −14.78117834613002, −13.70247582198175, −13.23601230594621, −12.55553078766583, −11.56704924736971, −10.70523564882565, −10.18658571215549, −9.310116363210926, −8.822830716811895, −7.869311668851587, −6.739307806272696, −6.301716425281838, −4.800027621823474, −4.555065390086967, −2.886815654699188, −1.441032060822410, 0,
1.441032060822410, 2.886815654699188, 4.555065390086967, 4.800027621823474, 6.301716425281838, 6.739307806272696, 7.869311668851587, 8.822830716811895, 9.310116363210926, 10.18658571215549, 10.70523564882565, 11.56704924736971, 12.55553078766583, 13.23601230594621, 13.70247582198175, 14.78117834613002, 15.43500679266512, 16.41093970400270, 17.05626771105392, 17.42174941409241, 18.01599762312436, 18.80154326137500, 19.40426436288095, 19.88527390088876
