| L(s) = 1 | − 2·2-s + 2·4-s + 5-s + 7-s − 2·10-s + 11-s + 13-s − 2·14-s − 4·16-s + 19-s + 2·20-s − 2·22-s − 3·23-s − 4·25-s − 2·26-s + 2·28-s − 2·29-s + 8·32-s + 35-s + 6·37-s − 2·38-s − 41-s + 5·43-s + 2·44-s + 6·46-s − 12·47-s + 49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s + 0.447·5-s + 0.377·7-s − 0.632·10-s + 0.301·11-s + 0.277·13-s − 0.534·14-s − 16-s + 0.229·19-s + 0.447·20-s − 0.426·22-s − 0.625·23-s − 4/5·25-s − 0.392·26-s + 0.377·28-s − 0.371·29-s + 1.41·32-s + 0.169·35-s + 0.986·37-s − 0.324·38-s − 0.156·41-s + 0.762·43-s + 0.301·44-s + 0.884·46-s − 1.75·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18207 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 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 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 16 T + p T^{2} \) |
| 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
−16.27733521059200, −15.71646359319649, −15.11458940615957, −14.39228181989489, −13.93960733734659, −13.33718435649492, −12.79014063349970, −11.93933582870414, −11.34115267695040, −11.08619286724719, −10.14595105007049, −9.925421412220595, −9.344355971458100, −8.717447143909823, −8.263348657057438, −7.583892163353573, −7.209359276698433, −6.242382991290555, −5.914260441062576, −4.913668886722960, −4.287244319906367, −3.429518544094301, −2.390370008091537, −1.756968636309207, −1.076274622030772, 0,
1.076274622030772, 1.756968636309207, 2.390370008091537, 3.429518544094301, 4.287244319906367, 4.913668886722960, 5.914260441062576, 6.242382991290555, 7.209359276698433, 7.583892163353573, 8.263348657057438, 8.717447143909823, 9.344355971458100, 9.925421412220595, 10.14595105007049, 11.08619286724719, 11.34115267695040, 11.93933582870414, 12.79014063349970, 13.33718435649492, 13.93960733734659, 14.39228181989489, 15.11458940615957, 15.71646359319649, 16.27733521059200
