L(s) = 1 | + 2·2-s + 3-s + 2·4-s − 3·5-s + 2·6-s − 7-s + 9-s − 6·10-s + 3·11-s + 2·12-s + 4·13-s − 2·14-s − 3·15-s − 4·16-s + 2·17-s + 2·18-s + 19-s − 6·20-s − 21-s + 6·22-s − 4·23-s + 4·25-s + 8·26-s + 27-s − 2·28-s + 5·29-s − 6·30-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 0.577·3-s + 4-s − 1.34·5-s + 0.816·6-s − 0.377·7-s + 1/3·9-s − 1.89·10-s + 0.904·11-s + 0.577·12-s + 1.10·13-s − 0.534·14-s − 0.774·15-s − 16-s + 0.485·17-s + 0.471·18-s + 0.229·19-s − 1.34·20-s − 0.218·21-s + 1.27·22-s − 0.834·23-s + 4/5·25-s + 1.56·26-s + 0.192·27-s − 0.377·28-s + 0.928·29-s − 1.09·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58989 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58989 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 53 | \( 1 \) |
good | 2 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−14.63242781623966, −13.99478610550360, −13.71245654666990, −13.11555132635807, −12.54664198755771, −12.16986571959658, −11.75237755562248, −11.29290311177753, −10.75040838880676, −10.00823905443932, −9.368344749208257, −8.747630302983893, −8.420814682739028, −7.700685859014811, −7.206660952929585, −6.585071609713177, −6.103541802959430, −5.539905534545749, −4.688744239152074, −4.212749474585935, −3.745455708041625, −3.394029198140307, −2.906002356880836, −1.924246699820907, −1.080186566810404, 0,
1.080186566810404, 1.924246699820907, 2.906002356880836, 3.394029198140307, 3.745455708041625, 4.212749474585935, 4.688744239152074, 5.539905534545749, 6.103541802959430, 6.585071609713177, 7.206660952929585, 7.700685859014811, 8.420814682739028, 8.747630302983893, 9.368344749208257, 10.00823905443932, 10.75040838880676, 11.29290311177753, 11.75237755562248, 12.16986571959658, 12.54664198755771, 13.11555132635807, 13.71245654666990, 13.99478610550360, 14.63242781623966