| L(s) = 1 | + 2-s − 4-s + 2·7-s − 3·8-s − 13-s + 2·14-s − 16-s − 5·17-s + 6·19-s + 2·23-s − 26-s − 2·28-s − 9·29-s − 2·31-s + 5·32-s − 5·34-s + 3·37-s + 6·38-s + 5·41-s + 2·46-s + 2·47-s − 3·49-s + 52-s + 9·53-s − 6·56-s − 9·58-s − 8·59-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s + 0.755·7-s − 1.06·8-s − 0.277·13-s + 0.534·14-s − 1/4·16-s − 1.21·17-s + 1.37·19-s + 0.417·23-s − 0.196·26-s − 0.377·28-s − 1.67·29-s − 0.359·31-s + 0.883·32-s − 0.857·34-s + 0.493·37-s + 0.973·38-s + 0.780·41-s + 0.294·46-s + 0.291·47-s − 3/7·49-s + 0.138·52-s + 1.23·53-s − 0.801·56-s − 1.18·58-s − 1.04·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 9 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 - 13 T + p T^{2} \) |
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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
−15.33639767499556, −14.76369317186453, −14.57425187152572, −13.88020455674034, −13.34994704214600, −13.06924076339492, −12.41205971966883, −11.73903267234141, −11.39265821006843, −10.85788338885826, −10.06764172728141, −9.388477996714892, −9.037679388533825, −8.510464866878081, −7.581584867504204, −7.400267467339825, −6.444673898703467, −5.782368747238695, −5.262713102943977, −4.746717219092465, −4.157997205956302, −3.543678240910162, −2.775859741483925, −2.022021327486827, −1.049612653249952, 0,
1.049612653249952, 2.022021327486827, 2.775859741483925, 3.543678240910162, 4.157997205956302, 4.746717219092465, 5.262713102943977, 5.782368747238695, 6.444673898703467, 7.400267467339825, 7.581584867504204, 8.510464866878081, 9.037679388533825, 9.388477996714892, 10.06764172728141, 10.85788338885826, 11.39265821006843, 11.73903267234141, 12.41205971966883, 13.06924076339492, 13.34994704214600, 13.88020455674034, 14.57425187152572, 14.76369317186453, 15.33639767499556
