| L(s) = 1 | − 2·4-s + 5-s + 7-s + 4·16-s − 3·17-s − 2·19-s − 2·20-s + 6·23-s − 4·25-s − 2·28-s + 29-s − 8·31-s + 35-s + 2·37-s − 2·41-s − 43-s + 3·47-s + 49-s + 4·53-s − 3·59-s − 4·61-s − 8·64-s − 7·67-s + 6·68-s + 2·71-s − 2·73-s + 4·76-s + ⋯ |
| L(s) = 1 | − 4-s + 0.447·5-s + 0.377·7-s + 16-s − 0.727·17-s − 0.458·19-s − 0.447·20-s + 1.25·23-s − 4/5·25-s − 0.377·28-s + 0.185·29-s − 1.43·31-s + 0.169·35-s + 0.328·37-s − 0.312·41-s − 0.152·43-s + 0.437·47-s + 1/7·49-s + 0.549·53-s − 0.390·59-s − 0.512·61-s − 64-s − 0.855·67-s + 0.727·68-s + 0.237·71-s − 0.234·73-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−13.15069887797919, −12.95582334933602, −12.36541826925890, −11.84776804095417, −11.25660726520234, −10.84638311438886, −10.39155682349026, −9.885692734829016, −9.319140448356058, −9.036395030621475, −8.592327027454161, −8.131717734974712, −7.491773541133213, −7.091976941058526, −6.473139758292258, −5.783989233147699, −5.523360687972517, −4.930961349984642, −4.367325046889638, −4.081646912284075, −3.307748297272716, −2.806315687443138, −1.969678767403111, −1.548419400987380, −0.7185593529199767, 0,
0.7185593529199767, 1.548419400987380, 1.969678767403111, 2.806315687443138, 3.307748297272716, 4.081646912284075, 4.367325046889638, 4.930961349984642, 5.523360687972517, 5.783989233147699, 6.473139758292258, 7.091976941058526, 7.491773541133213, 8.131717734974712, 8.592327027454161, 9.036395030621475, 9.319140448356058, 9.885692734829016, 10.39155682349026, 10.84638311438886, 11.25660726520234, 11.84776804095417, 12.36541826925890, 12.95582334933602, 13.15069887797919
