| L(s) = 1 | + 2·2-s + 4·4-s − 7.07·5-s + 8·8-s − 14.1·10-s − 40·11-s + 63.6·13-s + 16·16-s + 1.41·17-s − 11.3·19-s − 28.2·20-s − 80·22-s − 68·23-s − 75·25-s + 127.·26-s − 110·29-s + 118.·31-s + 32·32-s + 2.82·34-s − 20·37-s − 22.6·38-s − 56.5·40-s − 49.4·41-s − 340·43-s − 160·44-s − 136·46-s + 90.5·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.632·5-s + 0.353·8-s − 0.447·10-s − 1.09·11-s + 1.35·13-s + 0.250·16-s + 0.0201·17-s − 0.136·19-s − 0.316·20-s − 0.775·22-s − 0.616·23-s − 0.599·25-s + 0.960·26-s − 0.704·29-s + 0.688·31-s + 0.176·32-s + 0.0142·34-s − 0.0888·37-s − 0.0965·38-s − 0.223·40-s − 0.188·41-s − 1.20·43-s − 0.548·44-s − 0.435·46-s + 0.280·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 7.07T + 125T^{2} \) |
| 11 | \( 1 + 40T + 1.33e3T^{2} \) |
| 13 | \( 1 - 63.6T + 2.19e3T^{2} \) |
| 17 | \( 1 - 1.41T + 4.91e3T^{2} \) |
| 19 | \( 1 + 11.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 68T + 1.21e4T^{2} \) |
| 29 | \( 1 + 110T + 2.43e4T^{2} \) |
| 31 | \( 1 - 118.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 20T + 5.06e4T^{2} \) |
| 41 | \( 1 + 49.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 340T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 628T + 1.48e5T^{2} \) |
| 59 | \( 1 + 876.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 917.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 540T + 3.00e5T^{2} \) |
| 71 | \( 1 - 420T + 3.57e5T^{2} \) |
| 73 | \( 1 + 289.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 760T + 4.93e5T^{2} \) |
| 83 | \( 1 - 944.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.15e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 502.T + 9.12e5T^{2} \) |
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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
−9.344291069557412162470974285445, −8.134114786299807859530940129162, −7.79553157910202937552323822878, −6.55567738434376934896898272451, −5.80289680844946262702665755637, −4.79739450911468622155658371538, −3.85198287833411627621161539920, −3.01618959758702576407865535448, −1.66156170822587452417560838736, 0,
1.66156170822587452417560838736, 3.01618959758702576407865535448, 3.85198287833411627621161539920, 4.79739450911468622155658371538, 5.80289680844946262702665755637, 6.55567738434376934896898272451, 7.79553157910202937552323822878, 8.134114786299807859530940129162, 9.344291069557412162470974285445
