| L(s) = 1 | − 4·7-s − 2·11-s + 13-s − 4·17-s + 2·19-s − 2·23-s − 6·29-s − 8·31-s − 6·37-s − 10·41-s − 4·43-s + 9·49-s + 6·53-s + 6·59-s + 2·61-s − 4·67-s − 12·71-s + 2·73-s + 8·77-s − 8·79-s + 12·83-s − 14·89-s − 4·91-s − 10·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.603·11-s + 0.277·13-s − 0.970·17-s + 0.458·19-s − 0.417·23-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 1.56·41-s − 0.609·43-s + 9/7·49-s + 0.824·53-s + 0.781·59-s + 0.256·61-s − 0.488·67-s − 1.42·71-s + 0.234·73-s + 0.911·77-s − 0.900·79-s + 1.31·83-s − 1.48·89-s − 0.419·91-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.19844645790894, −14.70765963272777, −13.90887682275256, −13.43683687117501, −13.05924329795892, −12.78408271986622, −11.94180839653617, −11.66046129197330, −10.78173096974000, −10.50691271858265, −9.849154623337592, −9.427615149256613, −8.855774823861581, −8.401592138963335, −7.606612222636670, −6.915330054294903, −6.803594783702234, −5.886199175887072, −5.534447819409623, −4.868113342524662, −3.889546774583695, −3.603228970887885, −2.904748199836488, −2.207729515411550, −1.444346615948420, 0, 0,
1.444346615948420, 2.207729515411550, 2.904748199836488, 3.603228970887885, 3.889546774583695, 4.868113342524662, 5.534447819409623, 5.886199175887072, 6.803594783702234, 6.915330054294903, 7.606612222636670, 8.401592138963335, 8.855774823861581, 9.427615149256613, 9.849154623337592, 10.50691271858265, 10.78173096974000, 11.66046129197330, 11.94180839653617, 12.78408271986622, 13.05924329795892, 13.43683687117501, 13.90887682275256, 14.70765963272777, 15.19844645790894
