| L(s) = 1 | − 3-s + 5-s − 3·7-s + 9-s − 11-s − 13-s − 15-s + 3·17-s − 2·19-s + 3·21-s + 23-s − 4·25-s − 27-s − 29-s − 5·31-s + 33-s − 3·35-s + 3·37-s + 39-s + 12·41-s − 10·43-s + 45-s + 9·47-s + 2·49-s − 3·51-s − 6·53-s − 55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.13·7-s + 1/3·9-s − 0.301·11-s − 0.277·13-s − 0.258·15-s + 0.727·17-s − 0.458·19-s + 0.654·21-s + 0.208·23-s − 4/5·25-s − 0.192·27-s − 0.185·29-s − 0.898·31-s + 0.174·33-s − 0.507·35-s + 0.493·37-s + 0.160·39-s + 1.87·41-s − 1.52·43-s + 0.149·45-s + 1.31·47-s + 2/7·49-s − 0.420·51-s − 0.824·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8560667399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8560667399\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.61735273700976, −13.97405236890251, −13.38074083829419, −13.00359713530906, −12.50362899607922, −12.15217475676030, −11.42838153044386, −10.88817280743733, −10.40230513480070, −9.807307617431419, −9.518891678635422, −8.991757728594304, −8.167148575945851, −7.557413623055575, −7.088763181304879, −6.424254129388141, −5.879103876917851, −5.620003434960869, −4.828521291572976, −4.150449317990811, −3.501341043661607, −2.841477363965646, −2.151238982475771, −1.316362323789054, −0.3456568141484327,
0.3456568141484327, 1.316362323789054, 2.151238982475771, 2.841477363965646, 3.501341043661607, 4.150449317990811, 4.828521291572976, 5.620003434960869, 5.879103876917851, 6.424254129388141, 7.088763181304879, 7.557413623055575, 8.167148575945851, 8.991757728594304, 9.518891678635422, 9.807307617431419, 10.40230513480070, 10.88817280743733, 11.42838153044386, 12.15217475676030, 12.50362899607922, 13.00359713530906, 13.38074083829419, 13.97405236890251, 14.61735273700976
