L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (1 + 1.73i)9-s + (1.5 − 2.59i)11-s + 6·13-s − 0.999·15-s + (−2.5 + 4.33i)17-s + (−0.5 − 0.866i)19-s + (−3.5 − 6.06i)23-s + (2 − 3.46i)25-s + 5·27-s + 2·29-s + (2.5 − 4.33i)31-s + (−1.5 − 2.59i)33-s + (−1.5 − 2.59i)37-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (0.333 + 0.577i)9-s + (0.452 − 0.783i)11-s + 1.66·13-s − 0.258·15-s + (−0.606 + 1.05i)17-s + (−0.114 − 0.198i)19-s + (−0.729 − 1.26i)23-s + (0.400 − 0.692i)25-s + 0.962·27-s + 0.371·29-s + (0.449 − 0.777i)31-s + (−0.261 − 0.452i)33-s + (−0.246 − 0.427i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.59529 - 0.790832i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59529 - 0.790832i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.5 + 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.5 + 6.06i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + (-2.5 + 4.33i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (2.5 + 4.33i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.5 - 12.9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.5 + 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (4.5 - 7.79i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 - 0.866i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-3.5 - 6.06i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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
−10.45120110997570400713313408996, −8.921887815409750949620677954362, −8.490303029741058947845735969183, −7.83561505589564766315194885764, −6.50564176322530112745327498520, −6.05334164866168513289386213102, −4.55801081943033676285644279871, −3.77804885485341814298603916660, −2.32652803219185247584283719886, −1.03981421988973051146542624292,
1.44367073355545240069413480732, 3.13856635460184141872895251863, 3.88577616411109106626553667471, 4.83713227754517657544707365630, 6.19609291265192685697859777783, 6.88545609067547481526361064753, 7.86389071837176003732897133425, 8.975966674846728210555481551485, 9.439996242550285762224164905370, 10.38035833207847558530465975961