L(s) = 1 | + (0.760 + 1.31i)2-s + (−1.06 − 1.84i)3-s + (−0.156 + 0.270i)4-s − 0.589·5-s + (1.61 − 2.80i)6-s + 2.56·8-s + (−0.760 + 1.31i)9-s + (−0.448 − 0.776i)10-s + (−0.760 − 1.31i)11-s + 0.664·12-s + (−3.32 − 1.39i)13-s + (0.626 + 1.08i)15-s + (2.26 + 3.92i)16-s + (2.39 − 4.15i)17-s − 2.31·18-s + (0.841 − 1.45i)19-s + ⋯ |
L(s) = 1 | + (0.537 + 0.931i)2-s + (−0.613 − 1.06i)3-s + (−0.0781 + 0.135i)4-s − 0.263·5-s + (0.660 − 1.14i)6-s + 0.907·8-s + (−0.253 + 0.439i)9-s + (−0.141 − 0.245i)10-s + (−0.229 − 0.397i)11-s + 0.191·12-s + (−0.922 − 0.386i)13-s + (0.161 + 0.280i)15-s + (0.565 + 0.980i)16-s + (0.581 − 1.00i)17-s − 0.545·18-s + (0.193 − 0.334i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.384 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.384 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19100 - 0.793666i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19100 - 0.793666i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.32 + 1.39i)T \) |
good | 2 | \( 1 + (-0.760 - 1.31i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (1.06 + 1.84i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + 0.589T + 5T^{2} \) |
| 11 | \( 1 + (0.760 + 1.31i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.39 + 4.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.841 + 1.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.886 + 1.53i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.44 + 5.96i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.08T + 31T^{2} \) |
| 37 | \( 1 + (0.704 + 1.22i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.677 - 1.17i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.77 + 10.0i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 0.464T + 47T^{2} \) |
| 53 | \( 1 - 8.24T + 53T^{2} \) |
| 59 | \( 1 + (5.93 - 10.2i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.24 + 2.14i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.78 - 6.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.30 - 5.71i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 16.3T + 73T^{2} \) |
| 79 | \( 1 + 14.9T + 79T^{2} \) |
| 83 | \( 1 + 10.1T + 83T^{2} \) |
| 89 | \( 1 + (-8.24 - 14.2i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.486 - 0.843i)T + (-48.5 - 84.0i)T^{2} \) |
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
−10.49876496323365772648126235188, −9.586035055152074136869054940301, −8.098082563735665238727703414560, −7.44585344208448070796575038553, −6.90745429644619070007091050113, −5.85667906557422222491843185524, −5.36054392558397897088730336285, −4.14251011538571415330961171129, −2.40507548721078497229187371039, −0.71053457165757075059635209742,
1.87272041631664334366659867596, 3.29057157443323716982563814079, 4.18959726849041110709906162794, 4.85252188583718203242907420173, 5.83117510110445585152383349944, 7.28766277332740391763409173554, 8.072086312271248416643421007843, 9.590834576514112787267966791358, 10.07530471665828623844191563603, 10.83792271760549351377683655653