| L(s) = 1 | − 2-s − 4-s − 4·5-s + 3·8-s + 4·10-s − 2·11-s − 13-s − 16-s − 6·17-s − 4·19-s + 4·20-s + 2·22-s − 6·23-s + 11·25-s + 26-s − 2·29-s − 3·31-s − 5·32-s + 6·34-s + 3·37-s + 4·38-s − 12·40-s − 2·41-s − 43-s + 2·44-s + 6·46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s − 1.78·5-s + 1.06·8-s + 1.26·10-s − 0.603·11-s − 0.277·13-s − 1/4·16-s − 1.45·17-s − 0.917·19-s + 0.894·20-s + 0.426·22-s − 1.25·23-s + 11/5·25-s + 0.196·26-s − 0.371·29-s − 0.538·31-s − 0.883·32-s + 1.02·34-s + 0.493·37-s + 0.648·38-s − 1.89·40-s − 0.312·41-s − 0.152·43-s + 0.301·44-s + 0.884·46-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
| \(L(1)\) |
\(\approx\) |
\(0.2659071138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2659071138\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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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
−9.483118808498740468161112352072, −8.591393144440081216186674262878, −8.174842458635384979825291824413, −7.47906214811694943254418977697, −6.69301809340706125234074868915, −5.22174626816378645292658033645, −4.26459885949450872653953319987, −3.85297585812455508834915893838, −2.27467032412786158333398018626, −0.40116959550324554090187768272,
0.40116959550324554090187768272, 2.27467032412786158333398018626, 3.85297585812455508834915893838, 4.26459885949450872653953319987, 5.22174626816378645292658033645, 6.69301809340706125234074868915, 7.47906214811694943254418977697, 8.174842458635384979825291824413, 8.591393144440081216186674262878, 9.483118808498740468161112352072
