L(s) = 1 | + (−1.31 + 0.761i)2-s + (1.49 − 2.59i)3-s + (0.160 − 0.277i)4-s + (−2.55 + 1.47i)5-s + 4.56i·6-s − 2.55i·8-s + (−2.98 − 5.17i)9-s + (2.24 − 3.88i)10-s + (2.05 + 1.18i)11-s + (−0.480 − 0.831i)12-s + (−2.32 − 2.75i)13-s + 8.82i·15-s + (2.26 + 3.93i)16-s + (−2.68 + 4.65i)17-s + (7.87 + 4.54i)18-s + (−4.63 + 2.67i)19-s + ⋯ |
L(s) = 1 | + (−0.932 + 0.538i)2-s + (0.864 − 1.49i)3-s + (0.0801 − 0.138i)4-s + (−1.14 + 0.658i)5-s + 1.86i·6-s − 0.904i·8-s + (−0.995 − 1.72i)9-s + (0.709 − 1.22i)10-s + (0.619 + 0.357i)11-s + (−0.138 − 0.240i)12-s + (−0.645 − 0.763i)13-s + 2.27i·15-s + (0.567 + 0.982i)16-s + (−0.651 + 1.12i)17-s + (1.85 + 1.07i)18-s + (−1.06 + 0.614i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.855 - 0.518i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0352432 + 0.126137i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0352432 + 0.126137i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (2.32 + 2.75i)T \) |
good | 2 | \( 1 + (1.31 - 0.761i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.49 + 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.55 - 1.47i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.05 - 1.18i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.68 - 4.65i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.63 - 2.67i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.39 - 2.41i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 0.585T + 29T^{2} \) |
| 31 | \( 1 + (7.21 + 4.16i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.585 + 0.337i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.2iT - 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + (0.465 - 0.268i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.95 - 6.84i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.32 + 3.07i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.39 + 2.41i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.64 - 2.10i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 1.25iT - 71T^{2} \) |
| 73 | \( 1 + (-9.40 - 5.43i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.14 + 5.44i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 1.74iT - 83T^{2} \) |
| 89 | \( 1 + (8.86 - 5.11i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 9.90iT - 97T^{2} \) |
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
−10.90573028409457349771791556282, −9.744847265946908769051815768476, −8.770001908049390720027989911980, −8.008963709710055093018707208089, −7.71128828677160508374612866576, −6.86382764833778120405177061399, −6.29093355918243117619486312964, −4.05490805956521754279191936260, −3.16042514878099686766922819577, −1.67830343170457241254821187881,
0.085395496088323932036127400653, 2.25085248763119429025874432587, 3.52619441812530238168188435003, 4.52902368153434017037729271916, 5.03791807145877929341731510052, 7.02197473604083284215472965704, 8.225796063316820588673286843671, 8.914729499159470788759718524985, 9.103952320186661034107969121993, 10.04190218528312069485855013859