| L(s) = 1 | − 2-s + 2·3-s + 4-s − 5-s − 2·6-s + 7-s − 8-s + 9-s + 10-s − 11-s + 2·12-s + 2·13-s − 14-s − 2·15-s + 16-s + 2·17-s − 18-s + 6·19-s − 20-s + 2·21-s + 22-s + 6·23-s − 2·24-s + 25-s − 2·26-s − 4·27-s + 28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s − 0.816·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.577·12-s + 0.554·13-s − 0.267·14-s − 0.516·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s + 1.37·19-s − 0.223·20-s + 0.436·21-s + 0.213·22-s + 1.25·23-s − 0.408·24-s + 1/5·25-s − 0.392·26-s − 0.769·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.630504771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.630504771\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−10.10222677050510003671062028867, −9.253271582702521562348222427541, −8.640640021382486068538034692354, −7.80682046403378123605554461083, −7.39856421390674101016288192849, −6.06140846296057676016307473445, −4.82038769623350485466626190745, −3.43318839022200470842040938315, −2.73795458670056035674196653301, −1.22384148583299311271745764464,
1.22384148583299311271745764464, 2.73795458670056035674196653301, 3.43318839022200470842040938315, 4.82038769623350485466626190745, 6.06140846296057676016307473445, 7.39856421390674101016288192849, 7.80682046403378123605554461083, 8.640640021382486068538034692354, 9.253271582702521562348222427541, 10.10222677050510003671062028867
