| L(s) = 1 | − 3-s − 5-s + 9-s + 4·11-s − 13-s + 15-s + 17-s − 4·19-s + 25-s − 27-s + 2·29-s − 8·31-s − 4·33-s + 10·37-s + 39-s + 10·41-s + 4·43-s − 45-s + 8·47-s − 7·49-s − 51-s + 2·53-s − 4·55-s + 4·57-s − 4·59-s − 6·61-s + 65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.20·11-s − 0.277·13-s + 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 1.43·31-s − 0.696·33-s + 1.64·37-s + 0.160·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s + 1.16·47-s − 49-s − 0.140·51-s + 0.274·53-s − 0.539·55-s + 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.124·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 212160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 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
−13.06143754945557, −12.63111369918151, −12.38606477398581, −11.74311908385755, −11.45040612767096, −10.91444648177517, −10.63521115696453, −9.951190380157932, −9.441761921041316, −9.057697539292533, −8.607148403237151, −7.933527847958757, −7.372445989070002, −7.167203461272248, −6.430102760049013, −5.929312563208100, −5.763947533942305, −4.823731653522194, −4.263625998944767, −4.187350586822668, −3.390841530832172, −2.762147703257540, −2.065590074735775, −1.367412982514577, −0.7816634748897731, 0,
0.7816634748897731, 1.367412982514577, 2.065590074735775, 2.762147703257540, 3.390841530832172, 4.187350586822668, 4.263625998944767, 4.823731653522194, 5.763947533942305, 5.929312563208100, 6.430102760049013, 7.167203461272248, 7.372445989070002, 7.933527847958757, 8.607148403237151, 9.057697539292533, 9.441761921041316, 9.951190380157932, 10.63521115696453, 10.91444648177517, 11.45040612767096, 11.74311908385755, 12.38606477398581, 12.63111369918151, 13.06143754945557
