L(s) = 1 | + 3-s − 5-s + 9-s + 4·11-s + 13-s − 15-s − 2·17-s − 4·19-s − 8·23-s + 25-s + 27-s − 6·29-s + 4·33-s − 10·37-s + 39-s + 6·41-s + 4·43-s − 45-s − 7·49-s − 2·51-s + 6·53-s − 4·55-s − 4·57-s − 12·59-s − 10·61-s − 65-s + 12·67-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 − 0.485·17-s − 0.917·19-s − 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s + 0.696·33-s − 1.64·37-s + 0.160·39-s + 0.937·41-s + 0.609·43-s − 0.149·45-s − 49-s − 0.280·51-s + 0.824·53-s − 0.539·55-s − 0.529·57-s − 1.56·59-s − 1.28·61-s − 0.124·65-s + 1.46·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.79550765367496977687020548132, −7.00903940754672323846858659615, −6.38088145409704739626610229536, −5.67755399015807948091554891992, −4.47894850307774153951749978228, −4.00669383580325092419691444239, −3.38466759968687583933288431637, −2.22587623792251497835702909802, −1.49926292764207715503827482994, 0,
1.49926292764207715503827482994, 2.22587623792251497835702909802, 3.38466759968687583933288431637, 4.00669383580325092419691444239, 4.47894850307774153951749978228, 5.67755399015807948091554891992, 6.38088145409704739626610229536, 7.00903940754672323846858659615, 7.79550765367496977687020548132