| L(s) = 1 | − 5-s − 6·13-s + 3·17-s + 3·23-s + 25-s + 6·29-s − 7·31-s − 8·37-s − 2·41-s + 10·43-s − 3·47-s − 7·49-s − 3·53-s + 2·59-s + 7·61-s + 6·65-s − 12·67-s + 12·71-s + 4·73-s + 3·79-s − 3·85-s + 12·89-s + 16·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.66·13-s + 0.727·17-s + 0.625·23-s + 1/5·25-s + 1.11·29-s − 1.25·31-s − 1.31·37-s − 0.312·41-s + 1.52·43-s − 0.437·47-s − 49-s − 0.412·53-s + 0.260·59-s + 0.896·61-s + 0.744·65-s − 1.46·67-s + 1.42·71-s + 0.468·73-s + 0.337·79-s − 0.325·85-s + 1.27·89-s + 1.62·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
| 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
−14.83650180923885, −14.46027450735639, −14.11777857801313, −13.34169978141502, −12.73758601894660, −12.28169004319922, −12.02326986398824, −11.32209848282281, −10.75688550909411, −10.24871618568575, −9.691386245060826, −9.205800719243855, −8.599710445750199, −7.926277174926242, −7.478530090140902, −7.012492912202460, −6.434968595287026, −5.597752173011678, −5.000840534722162, −4.701354082599778, −3.777961036191662, −3.245718094267601, −2.554364354015697, −1.849439112232893, −0.8700617026981721, 0,
0.8700617026981721, 1.849439112232893, 2.554364354015697, 3.245718094267601, 3.777961036191662, 4.701354082599778, 5.000840534722162, 5.597752173011678, 6.434968595287026, 7.012492912202460, 7.478530090140902, 7.926277174926242, 8.599710445750199, 9.205800719243855, 9.691386245060826, 10.24871618568575, 10.75688550909411, 11.32209848282281, 12.02326986398824, 12.28169004319922, 12.73758601894660, 13.34169978141502, 14.11777857801313, 14.46027450735639, 14.83650180923885
