| L(s) = 1 | − 2·2-s − 4·4-s − 11·5-s − 7·7-s + 24·8-s + 22·10-s − 11·11-s − 5·13-s + 14·14-s − 16·16-s + 118·17-s − 105·19-s + 44·20-s + 22·22-s + 68·23-s − 4·25-s + 10·26-s + 28·28-s + 195·29-s + 214·31-s − 160·32-s − 236·34-s + 77·35-s + 33·37-s + 210·38-s − 264·40-s + 376·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 0.983·5-s − 0.377·7-s + 1.06·8-s + 0.695·10-s − 0.301·11-s − 0.106·13-s + 0.267·14-s − 1/4·16-s + 1.68·17-s − 1.26·19-s + 0.491·20-s + 0.213·22-s + 0.616·23-s − 0.0319·25-s + 0.0754·26-s + 0.188·28-s + 1.24·29-s + 1.23·31-s − 0.883·32-s − 1.19·34-s + 0.371·35-s + 0.146·37-s + 0.896·38-s − 1.04·40-s + 1.43·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 2 | \( 1 + p T + p^{3} T^{2} \) |
| 5 | \( 1 + 11 T + p^{3} T^{2} \) |
| 13 | \( 1 + 5 T + p^{3} T^{2} \) |
| 17 | \( 1 - 118 T + p^{3} T^{2} \) |
| 19 | \( 1 + 105 T + p^{3} T^{2} \) |
| 23 | \( 1 - 68 T + p^{3} T^{2} \) |
| 29 | \( 1 - 195 T + p^{3} T^{2} \) |
| 31 | \( 1 - 214 T + p^{3} T^{2} \) |
| 37 | \( 1 - 33 T + p^{3} T^{2} \) |
| 41 | \( 1 - 376 T + p^{3} T^{2} \) |
| 43 | \( 1 + 168 T + p^{3} T^{2} \) |
| 47 | \( 1 + 61 T + p^{3} T^{2} \) |
| 53 | \( 1 + 24 T + p^{3} T^{2} \) |
| 59 | \( 1 + 625 T + p^{3} T^{2} \) |
| 61 | \( 1 + 558 T + p^{3} T^{2} \) |
| 67 | \( 1 - 173 T + p^{3} T^{2} \) |
| 71 | \( 1 + 168 T + p^{3} T^{2} \) |
| 73 | \( 1 - 973 T + p^{3} T^{2} \) |
| 79 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1458 T + p^{3} T^{2} \) |
| 89 | \( 1 - 198 T + p^{3} T^{2} \) |
| 97 | \( 1 + 352 T + p^{3} T^{2} \) |
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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
−9.690080354283100428463328679529, −8.630725087549140185856852816193, −8.029033511105383098361807536901, −7.36376097778730294582595768946, −6.16131591365869911538356879012, −4.87520469019249353765328428154, −4.06318388236473030531951589967, −2.92755015703034847216578885872, −1.10133640584924975490442967894, 0,
1.10133640584924975490442967894, 2.92755015703034847216578885872, 4.06318388236473030531951589967, 4.87520469019249353765328428154, 6.16131591365869911538356879012, 7.36376097778730294582595768946, 8.029033511105383098361807536901, 8.630725087549140185856852816193, 9.690080354283100428463328679529
