| L(s) = 1 | + 2-s − 7·4-s − 36·7-s − 15·8-s − 11·11-s − 2·13-s − 36·14-s + 41·16-s + 66·17-s + 140·19-s − 11·22-s − 68·23-s − 2·26-s + 252·28-s − 150·29-s − 128·31-s + 161·32-s + 66·34-s + 314·37-s + 140·38-s + 118·41-s − 172·43-s + 77·44-s − 68·46-s − 324·47-s + 953·49-s + 14·52-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 1.94·7-s − 0.662·8-s − 0.301·11-s − 0.0426·13-s − 0.687·14-s + 0.640·16-s + 0.941·17-s + 1.69·19-s − 0.106·22-s − 0.616·23-s − 0.0150·26-s + 1.70·28-s − 0.960·29-s − 0.741·31-s + 0.889·32-s + 0.332·34-s + 1.39·37-s + 0.597·38-s + 0.449·41-s − 0.609·43-s + 0.263·44-s − 0.217·46-s − 1.00·47-s + 2.77·49-s + 0.0373·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 7 | \( 1 + 36 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 140 T + p^{3} T^{2} \) |
| 23 | \( 1 + 68 T + p^{3} T^{2} \) |
| 29 | \( 1 + 150 T + p^{3} T^{2} \) |
| 31 | \( 1 + 128 T + p^{3} T^{2} \) |
| 37 | \( 1 - 314 T + p^{3} T^{2} \) |
| 41 | \( 1 - 118 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 + 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 82 T + p^{3} T^{2} \) |
| 59 | \( 1 - 740 T + p^{3} T^{2} \) |
| 61 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 67 | \( 1 - 124 T + p^{3} T^{2} \) |
| 71 | \( 1 - 988 T + p^{3} T^{2} \) |
| 73 | \( 1 + 2 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1100 T + p^{3} T^{2} \) |
| 83 | \( 1 + 868 T + p^{3} T^{2} \) |
| 89 | \( 1 - 470 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1186 T + p^{3} T^{2} \) |
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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
−8.178439633100723224452543835970, −7.42670379657934850564193498228, −6.52921552504483365176771341530, −5.67463282166033818677777797230, −5.26314626977860154347703265061, −3.89124221607078229895311648271, −3.47594538731845498595350935005, −2.66218828846440034747551477276, −0.918352187082517849037441252772, 0,
0.918352187082517849037441252772, 2.66218828846440034747551477276, 3.47594538731845498595350935005, 3.89124221607078229895311648271, 5.26314626977860154347703265061, 5.67463282166033818677777797230, 6.52921552504483365176771341530, 7.42670379657934850564193498228, 8.178439633100723224452543835970
