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 | \( 1 - 2T + 97T^{2} \) |
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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
−10.38035833207847558530465975961, −9.439996242550285762224164905370, −8.975966674846728210555481551485, −7.86389071837176003732897133425, −6.88545609067547481526361064753, −6.19609291265192685697859777783, −4.83713227754517657544707365630, −3.88577616411109106626553667471, −3.13856635460184141872895251863, −1.44367073355545240069413480732,
1.03981421988973051146542624292, 2.32652803219185247584283719886, 3.77804885485341814298603916660, 4.55801081943033676285644279871, 6.05334164866168513289386213102, 6.50564176322530112745327498520, 7.83561505589564766315194885764, 8.490303029741058947845735969183, 8.921887815409750949620677954362, 10.45120110997570400713313408996