L(s) = 1 | − 2-s − 2·3-s + 4-s + 2·5-s + 2·6-s − 7-s − 8-s + 9-s − 2·10-s − 2·12-s + 2·13-s + 14-s − 4·15-s + 16-s − 18-s + 6·19-s + 2·20-s + 2·21-s + 2·24-s − 25-s − 2·26-s + 4·27-s − 28-s − 29-s + 4·30-s − 2·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.894·5-s + 0.816·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s − 0.577·12-s + 0.554·13-s + 0.267·14-s − 1.03·15-s + 1/4·16-s − 0.235·18-s + 1.37·19-s + 0.447·20-s + 0.436·21-s + 0.408·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.188·28-s − 0.185·29-s + 0.730·30-s − 0.359·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 214774 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 214774 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.220632487\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.220632487\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−12.87093833671667, −12.50546661833150, −11.84966413815345, −11.56173670526340, −11.05465418940902, −10.72327230755122, −10.10959272589229, −9.797451545252765, −9.352001610288131, −8.901316932026173, −8.295217338736514, −7.769660636990616, −7.110263793152435, −6.757746117317779, −6.251386271291874, −5.678273404598226, −5.518109728564083, −5.016027441976565, −4.140248733917519, −3.565422467434453, −2.894581896504852, −2.316623574285591, −1.587753460650570, −1.023681440125763, −0.4354147904749658,
0.4354147904749658, 1.023681440125763, 1.587753460650570, 2.316623574285591, 2.894581896504852, 3.565422467434453, 4.140248733917519, 5.016027441976565, 5.518109728564083, 5.678273404598226, 6.251386271291874, 6.757746117317779, 7.110263793152435, 7.769660636990616, 8.295217338736514, 8.901316932026173, 9.352001610288131, 9.797451545252765, 10.10959272589229, 10.72327230755122, 11.05465418940902, 11.56173670526340, 11.84966413815345, 12.50546661833150, 12.87093833671667