| L(s) = 1 | + (−2 − 3.46i)2-s + (−7.99 + 13.8i)4-s + (42 − 72.7i)5-s + (96.5 + 167. i)7-s + 63.9·8-s − 336·10-s + (174 + 301. i)11-s + (−422.5 + 731. i)13-s + (386 − 668. i)14-s + (−128 − 221. i)16-s − 1.69e3·17-s − 79·19-s + (672 + 1.16e3i)20-s + (696 − 1.20e3i)22-s + (−282 + 488. i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.751 − 1.30i)5-s + (0.744 + 1.28i)7-s + 0.353·8-s − 1.06·10-s + (0.433 + 0.750i)11-s + (−0.693 + 1.20i)13-s + (0.526 − 0.911i)14-s + (−0.125 − 0.216i)16-s − 1.41·17-s − 0.0502·19-s + (0.375 + 0.650i)20-s + (0.306 − 0.531i)22-s + (−0.111 + 0.192i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.502577135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.502577135\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2 + 3.46i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-42 + 72.7i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-96.5 - 167. i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-174 - 301. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 + (422.5 - 731. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + 1.69e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 79T + 2.47e6T^{2} \) |
| 23 | \( 1 + (282 - 488. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-3.21e3 - 5.57e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.47e3 - 4.27e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + 3.80e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (6.24e3 - 1.08e4i)T + (-5.79e7 - 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-2.46e3 - 4.27e3i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (-4.06e3 - 7.03e3i)T + (-1.14e8 + 1.98e8i)T^{2} \) |
| 53 | \( 1 - 3.31e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-2.12e4 + 3.67e4i)T + (-3.57e8 - 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-8.91e3 - 1.54e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-3.38e4 + 5.86e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + 2.81e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.39e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-4.19e4 - 7.27e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (1.66e4 + 2.89e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + 7.78e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-1.04e3 - 1.80e3i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
| 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
−12.15490745415653075997527975173, −11.26315466326530653463021020761, −9.756926164438218341017643151664, −8.993232413470233434922444012829, −8.543864177320033134863076134925, −6.79628945425255140066514759818, −5.19485153179162738957525793120, −4.51724769836258279719697298156, −2.22056691979253182518084928382, −1.54280496346752375803431624158,
0.55108011856373619595216830498, 2.36170283160482396916777249695, 4.05091563182818141220053281248, 5.60062560786406639146654172901, 6.72498161005839064641572487743, 7.44992788829986564092597309165, 8.612975127062494389332868233632, 10.11178330095123158429771520958, 10.53914852831916839699276723988, 11.46326366265977404668892553948
