| L(s) = 1 | + (0.0664 − 0.0383i)2-s + (−2.92 − 0.660i)3-s + (−1.99 + 3.45i)4-s + 4.07i·5-s + (−0.219 + 0.0684i)6-s + (−6.97 + 0.595i)7-s + 0.613i·8-s + (8.12 + 3.86i)9-s + (0.156 + 0.270i)10-s + 12.4i·11-s + (8.12 − 8.80i)12-s + (−5.97 − 10.3i)13-s + (−0.440 + 0.307i)14-s + (2.69 − 11.9i)15-s + (−7.96 − 13.7i)16-s + (14.2 − 8.24i)17-s + ⋯ |
| L(s) = 1 | + (0.0332 − 0.0191i)2-s + (−0.975 − 0.220i)3-s + (−0.499 + 0.864i)4-s + 0.815i·5-s + (−0.0366 + 0.0114i)6-s + (−0.996 + 0.0850i)7-s + 0.0766i·8-s + (0.903 + 0.429i)9-s + (0.0156 + 0.0270i)10-s + 1.13i·11-s + (0.677 − 0.733i)12-s + (−0.459 − 0.796i)13-s + (−0.0314 + 0.0219i)14-s + (0.179 − 0.795i)15-s + (−0.497 − 0.862i)16-s + (0.840 − 0.485i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.545 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.269493 + 0.497273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.269493 + 0.497273i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.92 + 0.660i)T \) |
| 7 | \( 1 + (6.97 - 0.595i)T \) |
| good | 2 | \( 1 + (-0.0664 + 0.0383i)T + (2 - 3.46i)T^{2} \) |
| 5 | \( 1 - 4.07iT - 25T^{2} \) |
| 11 | \( 1 - 12.4iT - 121T^{2} \) |
| 13 | \( 1 + (5.97 + 10.3i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (-14.2 + 8.24i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (3.69 - 6.39i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 - 27.9iT - 529T^{2} \) |
| 29 | \( 1 + (-32.2 - 18.5i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (16.2 - 28.1i)T + (-480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (10.3 - 17.9i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (26.1 - 15.0i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (12.7 - 22.1i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-68.4 + 39.5i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (61.5 - 35.5i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (37.3 + 21.5i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-11.2 - 19.4i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-43.0 + 74.6i)T + (-2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 102. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-0.403 - 0.699i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-13.5 - 23.4i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-36.4 - 21.0i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-36.1 - 20.8i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-6.66 + 11.5i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
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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
−15.24393841678684512856036430714, −13.75845001542822686094492674134, −12.53875543258654317230730651818, −12.13997062480767320358331881146, −10.51578956029609467015897140816, −9.599847199042161627083508872572, −7.61759677068337380191035326903, −6.75830482095282819616459736302, −5.07689049621570319676630843405, −3.22792206174740307699393402859,
0.59413969119455105786545332309, 4.26692191168672387312777512371, 5.57391262667737390778022705655, 6.55040198108333176437374397279, 8.790895504557156187105724241231, 9.815678255879855409240738339495, 10.79179444103019010285664581986, 12.19832787321939633214325362049, 13.11303988243103836299450020733, 14.28702425871908376255493182539
