L(s) = 1 | + 2-s − 3-s − 4-s + 5-s − 6-s + 7-s − 3·8-s + 9-s + 10-s − 11-s + 12-s − 2·13-s + 14-s − 15-s − 16-s − 6·17-s + 18-s + 4·19-s − 20-s − 21-s − 22-s − 4·23-s + 3·24-s + 25-s − 2·26-s − 27-s − 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 0.267·14-s − 0.258·15-s − 1/4·16-s − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.223·20-s − 0.218·21-s − 0.213·22-s − 0.834·23-s + 0.612·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
good | 2 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−9.438199599986511453147505381379, −8.688876376173663543847229264643, −7.63560441370956662338729532790, −6.62629853665671709769618392561, −5.77641410636928002092460840171, −5.03185863253334296879036362235, −4.42103096904459177558971112803, −3.24231752509856341716457202217, −1.90210262429212048684962532791, 0,
1.90210262429212048684962532791, 3.24231752509856341716457202217, 4.42103096904459177558971112803, 5.03185863253334296879036362235, 5.77641410636928002092460840171, 6.62629853665671709769618392561, 7.63560441370956662338729532790, 8.688876376173663543847229264643, 9.438199599986511453147505381379