| L(s) = 1 | − 2·2-s + 3-s + 2·4-s + 3·5-s − 2·6-s + 9-s − 6·10-s + 11-s + 2·12-s − 2·13-s + 3·15-s − 4·16-s + 17-s − 2·18-s + 19-s + 6·20-s − 2·22-s − 4·23-s + 4·25-s + 4·26-s + 27-s − 2·29-s − 6·30-s + 6·31-s + 8·32-s + 33-s − 2·34-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 0.577·3-s + 4-s + 1.34·5-s − 0.816·6-s + 1/3·9-s − 1.89·10-s + 0.301·11-s + 0.577·12-s − 0.554·13-s + 0.774·15-s − 16-s + 0.242·17-s − 0.471·18-s + 0.229·19-s + 1.34·20-s − 0.426·22-s − 0.834·23-s + 4/5·25-s + 0.784·26-s + 0.192·27-s − 0.371·29-s − 1.09·30-s + 1.07·31-s + 1.41·32-s + 0.174·33-s − 0.342·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2793 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.450120799\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.450120799\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.959163091125505196869204673189, −8.240904623337692395017705850814, −7.50551633645791376109359922005, −6.77741729329520342893941173390, −5.95880974340205553779958707744, −5.02464724211845145360498207900, −3.92167943470614491379352796776, −2.54192725156556063621144024653, −1.98299256211429317844363710869, −0.930874084897434565967858870729,
0.930874084897434565967858870729, 1.98299256211429317844363710869, 2.54192725156556063621144024653, 3.92167943470614491379352796776, 5.02464724211845145360498207900, 5.95880974340205553779958707744, 6.77741729329520342893941173390, 7.50551633645791376109359922005, 8.240904623337692395017705850814, 8.959163091125505196869204673189
