| L(s) = 1 | + 2-s − 4-s + 3·5-s − 3·8-s + 3·10-s − 5·11-s − 6·13-s − 16-s − 6·17-s − 3·19-s − 3·20-s − 5·22-s − 23-s + 4·25-s − 6·26-s + 2·29-s + 3·31-s + 5·32-s − 6·34-s + 3·37-s − 3·38-s − 9·40-s − 9·41-s + 6·43-s + 5·44-s − 46-s + 6·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.34·5-s − 1.06·8-s + 0.948·10-s − 1.50·11-s − 1.66·13-s − 1/4·16-s − 1.45·17-s − 0.688·19-s − 0.670·20-s − 1.06·22-s − 0.208·23-s + 4/5·25-s − 1.17·26-s + 0.371·29-s + 0.538·31-s + 0.883·32-s − 1.02·34-s + 0.493·37-s − 0.486·38-s − 1.42·40-s − 1.40·41-s + 0.914·43-s + 0.753·44-s − 0.147·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 + 12 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−9.265743020815139277667174323232, −8.600848877784761340952899741575, −7.51628200491268451704050962091, −6.47884273436117871307974498700, −5.68667112029587934302581219019, −4.97324351018055834490293383395, −4.37690537415494649135748951679, −2.73512468913067121254189272250, −2.26481617188159673197543529003, 0,
2.26481617188159673197543529003, 2.73512468913067121254189272250, 4.37690537415494649135748951679, 4.97324351018055834490293383395, 5.68667112029587934302581219019, 6.47884273436117871307974498700, 7.51628200491268451704050962091, 8.600848877784761340952899741575, 9.265743020815139277667174323232
