L(s) = 1 | + 2-s − 3·3-s + 4-s − 5-s − 3·6-s + 7-s + 8-s + 6·9-s − 10-s − 2·11-s − 3·12-s + 14-s + 3·15-s + 16-s − 3·17-s + 6·18-s − 8·19-s − 20-s − 3·21-s − 2·22-s − 4·23-s − 3·24-s − 4·25-s − 9·27-s + 28-s − 5·29-s + 3·30-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.447·5-s − 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s − 0.316·10-s − 0.603·11-s − 0.866·12-s + 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.727·17-s + 1.41·18-s − 1.83·19-s − 0.223·20-s − 0.654·21-s − 0.426·22-s − 0.834·23-s − 0.612·24-s − 4/5·25-s − 1.73·27-s + 0.188·28-s − 0.928·29-s + 0.547·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 574 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 41 | \( 1 + T \) |
good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 5 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−10.82348754711184792755959334111, −9.814547451875919314932987373145, −8.260721856204501813182654193846, −7.29964473624905066637087401045, −6.31811319083545584980505157205, −5.70227872805281049394495893274, −4.65757518643900400132339268063, −4.04251778571385567301847619669, −2.03110414024071717045287920222, 0,
2.03110414024071717045287920222, 4.04251778571385567301847619669, 4.65757518643900400132339268063, 5.70227872805281049394495893274, 6.31811319083545584980505157205, 7.29964473624905066637087401045, 8.260721856204501813182654193846, 9.814547451875919314932987373145, 10.82348754711184792755959334111