L(s) = 1 | − 3-s − 5-s + 7-s + 9-s + 4·11-s + 6·13-s + 15-s − 6·17-s + 6·19-s − 21-s − 23-s + 25-s − 27-s − 2·29-s − 8·31-s − 4·33-s − 35-s − 8·37-s − 6·39-s − 10·41-s − 4·43-s − 45-s − 12·47-s + 49-s + 6·51-s − 6·53-s − 4·55-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 1.66·13-s + 0.258·15-s − 1.45·17-s + 1.37·19-s − 0.218·21-s − 0.208·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s − 1.43·31-s − 0.696·33-s − 0.169·35-s − 1.31·37-s − 0.960·39-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.75·47-s + 1/7·49-s + 0.840·51-s − 0.824·53-s − 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−7.03006976309984419732306100212, −6.80743205754775227254413357550, −6.01057369855440498797083645690, −5.28786113739241214909370411714, −4.58694293825222626034596344591, −3.69423974735149489940934354479, −3.42865933473498536313227527018, −1.81416745756692951175793153535, −1.30534759960674675816245604323, 0,
1.30534759960674675816245604323, 1.81416745756692951175793153535, 3.42865933473498536313227527018, 3.69423974735149489940934354479, 4.58694293825222626034596344591, 5.28786113739241214909370411714, 6.01057369855440498797083645690, 6.80743205754775227254413357550, 7.03006976309984419732306100212