L(s) = 1 | + (0.173 + 0.300i)7-s + (1.26 − 0.223i)13-s + (0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.592 + 0.342i)31-s + 0.684i·37-s + (−1.43 − 1.20i)43-s + (0.439 − 0.761i)49-s + (−1.17 + 0.984i)61-s + (−0.673 + 1.85i)67-s + (0.266 − 1.50i)73-s + (−1.93 − 0.342i)79-s + (0.286 + 0.342i)91-s + (−0.592 − 1.62i)97-s + (−1.11 − 0.642i)103-s + ⋯ |
L(s) = 1 | + (0.173 + 0.300i)7-s + (1.26 − 0.223i)13-s + (0.5 + 0.866i)19-s + (−0.173 − 0.984i)25-s + (−0.592 + 0.342i)31-s + 0.684i·37-s + (−1.43 − 1.20i)43-s + (0.439 − 0.761i)49-s + (−1.17 + 0.984i)61-s + (−0.673 + 1.85i)67-s + (0.266 − 1.50i)73-s + (−1.93 − 0.342i)79-s + (0.286 + 0.342i)91-s + (−0.592 − 1.62i)97-s + (−1.11 − 0.642i)103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 - 0.120i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9934736495\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9934736495\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.26 + 0.223i)T + (0.939 - 0.342i)T^{2} \) |
| 17 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 23 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (0.592 - 0.342i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.684iT - T^{2} \) |
| 41 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (1.17 - 0.984i)T + (0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (0.673 - 1.85i)T + (-0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \) |
| 79 | \( 1 + (1.93 + 0.342i)T + (0.939 + 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 89 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (0.592 + 1.62i)T + (-0.766 + 0.642i)T^{2} \) |
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
−10.60826526278565807621625997636, −9.987108360232228153659880026543, −8.789692427325710914689404402639, −8.295767936786091469123624481403, −7.21501617772971098462621695038, −6.15445842382711140112471029810, −5.41229490700873079074749976410, −4.15150028260180327291670876234, −3.12840136531938841681373630438, −1.60769552276505894918811906902,
1.49219031819791772022830470178, 3.11232544877485692927436509502, 4.13736558675381106353360772261, 5.24878685478937538249821368770, 6.24709248013325533580793953248, 7.18468926952169906891068501084, 8.067705491979884392212325146904, 9.020117991550762485058564663836, 9.704030326828789823234205827512, 10.99292849380964429654084901097