L(s) = 1 | + 2-s + 3-s + 4-s + 2·5-s + 6-s + 2·7-s + 8-s + 9-s + 2·10-s − 11-s + 12-s + 2·14-s + 2·15-s + 16-s − 17-s + 18-s − 6·19-s + 2·20-s + 2·21-s − 22-s − 6·23-s + 24-s − 25-s + 27-s + 2·28-s − 2·29-s + 2·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.632·10-s − 0.301·11-s + 0.288·12-s + 0.534·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.37·19-s + 0.447·20-s + 0.436·21-s − 0.213·22-s − 1.25·23-s + 0.204·24-s − 1/5·25-s + 0.192·27-s + 0.377·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189618 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−13.28582320534647, −13.04599885275312, −12.63921529798052, −11.97817377303267, −11.49726381707720, −11.01851520965774, −10.57627243468486, −9.995170347266376, −9.713415430654421, −9.046699565852487, −8.506426462748760, −8.114249566612408, −7.601349626115677, −7.099791091246954, −6.420908282495581, −5.987982267221447, −5.604846211664618, −4.988041655057666, −4.351894087882593, −4.063179351509293, −3.439863157095992, −2.521842938308500, −2.253262293058802, −1.846216532620069, −1.115981504687644, 0,
1.115981504687644, 1.846216532620069, 2.253262293058802, 2.521842938308500, 3.439863157095992, 4.063179351509293, 4.351894087882593, 4.988041655057666, 5.604846211664618, 5.987982267221447, 6.420908282495581, 7.099791091246954, 7.601349626115677, 8.114249566612408, 8.506426462748760, 9.046699565852487, 9.713415430654421, 9.995170347266376, 10.57627243468486, 11.01851520965774, 11.49726381707720, 11.97817377303267, 12.63921529798052, 13.04599885275312, 13.28582320534647