| L(s) = 1 | − 2-s + 4-s − 5-s − 8-s + 10-s + 6·13-s + 16-s + 2·17-s − 19-s − 20-s + 25-s − 6·26-s − 32-s − 2·34-s − 2·37-s + 38-s + 40-s − 2·41-s + 4·43-s − 7·49-s − 50-s + 6·52-s + 2·53-s + 12·59-s + 10·61-s + 64-s − 6·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.353·8-s + 0.316·10-s + 1.66·13-s + 1/4·16-s + 0.485·17-s − 0.229·19-s − 0.223·20-s + 1/5·25-s − 1.17·26-s − 0.176·32-s − 0.342·34-s − 0.328·37-s + 0.162·38-s + 0.158·40-s − 0.312·41-s + 0.609·43-s − 49-s − 0.141·50-s + 0.832·52-s + 0.274·53-s + 1.56·59-s + 1.28·61-s + 1/8·64-s − 0.744·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206910 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.25225295743920, −12.80355348829121, −12.21161591079129, −11.71149301931078, −11.35109288573279, −10.95853665161256, −10.31801577835830, −10.17777903305152, −9.371219026460336, −8.984329838100636, −8.479914099597973, −8.162769083821703, −7.683768359812589, −7.058187947206336, −6.637261417161791, −6.121118580048343, −5.611539957640097, −5.087100327451933, −4.287800800856684, −3.783840728902064, −3.376326351401611, −2.719944414555086, −2.011230457568702, −1.314182687145238, −0.8588787612163877, 0,
0.8588787612163877, 1.314182687145238, 2.011230457568702, 2.719944414555086, 3.376326351401611, 3.783840728902064, 4.287800800856684, 5.087100327451933, 5.611539957640097, 6.121118580048343, 6.637261417161791, 7.058187947206336, 7.683768359812589, 8.162769083821703, 8.479914099597973, 8.984329838100636, 9.371219026460336, 10.17777903305152, 10.31801577835830, 10.95853665161256, 11.35109288573279, 11.71149301931078, 12.21161591079129, 12.80355348829121, 13.25225295743920
