| L(s) = 1 | + 2-s + 4-s + 5-s − 4·7-s + 8-s + 10-s − 6·11-s − 4·13-s − 4·14-s + 16-s − 6·17-s + 20-s − 6·22-s − 6·23-s + 25-s − 4·26-s − 4·28-s − 2·29-s + 32-s − 6·34-s − 4·35-s − 8·37-s + 40-s − 10·41-s − 4·43-s − 6·44-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 1.51·7-s + 0.353·8-s + 0.316·10-s − 1.80·11-s − 1.10·13-s − 1.06·14-s + 1/4·16-s − 1.45·17-s + 0.223·20-s − 1.27·22-s − 1.25·23-s + 1/5·25-s − 0.784·26-s − 0.755·28-s − 0.371·29-s + 0.176·32-s − 1.02·34-s − 0.676·35-s − 1.31·37-s + 0.158·40-s − 1.56·41-s − 0.609·43-s − 0.904·44-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−15.66272501197824, −15.05106999758859, −14.55193097127547, −13.67105799148168, −13.37815068640839, −13.15424721220935, −12.45160722939467, −12.14376181204897, −11.41005918847806, −10.57072261470681, −10.28970129536663, −9.863855854206191, −9.237480933016114, −8.520637085894012, −7.851494805501812, −7.198820455051753, −6.658124287886020, −6.254856896707411, −5.391702235636101, −5.149951177035239, −4.365219525719791, −3.576392302105762, −2.935301087227828, −2.386960016073573, −1.869341062158704, 0, 0,
1.869341062158704, 2.386960016073573, 2.935301087227828, 3.576392302105762, 4.365219525719791, 5.149951177035239, 5.391702235636101, 6.254856896707411, 6.658124287886020, 7.198820455051753, 7.851494805501812, 8.520637085894012, 9.237480933016114, 9.863855854206191, 10.28970129536663, 10.57072261470681, 11.41005918847806, 12.14376181204897, 12.45160722939467, 13.15424721220935, 13.37815068640839, 13.67105799148168, 14.55193097127547, 15.05106999758859, 15.66272501197824
