| L(s) = 1 | − 2·5-s − 2·13-s + 17-s − 25-s − 2·29-s + 8·31-s − 6·37-s − 6·41-s − 4·43-s − 14·53-s + 8·59-s − 14·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s − 8·79-s + 16·83-s − 2·85-s + 2·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.554·13-s + 0.242·17-s − 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 1.92·53-s + 1.04·59-s − 1.79·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s + 1.75·83-s − 0.216·85-s + 0.211·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5745529514\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5745529514\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.54699651980936, −13.12326283227918, −12.43694081429278, −12.05998801978936, −11.74185098958261, −11.26460887138689, −10.61771733040701, −10.20563929760815, −9.703099247909614, −9.159949685355082, −8.540599763091441, −8.136372769014131, −7.582591179979043, −7.299895013238835, −6.460762007314984, −6.261886562035607, −5.320581104246684, −4.925015391504844, −4.403090982212319, −3.747896478879083, −3.266935294852462, −2.695629258439918, −1.882127817260399, −1.236266447670572, −0.2383624658098372,
0.2383624658098372, 1.236266447670572, 1.882127817260399, 2.695629258439918, 3.266935294852462, 3.747896478879083, 4.403090982212319, 4.925015391504844, 5.320581104246684, 6.261886562035607, 6.460762007314984, 7.299895013238835, 7.582591179979043, 8.136372769014131, 8.540599763091441, 9.159949685355082, 9.703099247909614, 10.20563929760815, 10.61771733040701, 11.26460887138689, 11.74185098958261, 12.05998801978936, 12.43694081429278, 13.12326283227918, 13.54699651980936
