L(s) = 1 | + (5.31 − 9.19i)2-s + (−40.4 − 70.0i)4-s + (−33.7 + 58.4i)5-s + (−120. + 48.3i)7-s − 518.·8-s + (358. + 620. i)10-s + (−261. − 452. i)11-s + 76.6·13-s + (−193. + 1.36e3i)14-s + (−1.46e3 + 2.53e3i)16-s + (−634. − 1.09e3i)17-s + (946. − 1.63e3i)19-s + 5.45e3·20-s − 5.55e3·22-s + (575. − 995. i)23-s + ⋯ |
L(s) = 1 | + (0.938 − 1.62i)2-s + (−1.26 − 2.18i)4-s + (−0.603 + 1.04i)5-s + (−0.927 + 0.373i)7-s − 2.86·8-s + (1.13 + 1.96i)10-s + (−0.651 − 1.12i)11-s + 0.125·13-s + (−0.264 + 1.85i)14-s + (−1.42 + 2.47i)16-s + (−0.532 − 0.922i)17-s + (0.601 − 1.04i)19-s + 3.04·20-s − 2.44·22-s + (0.226 − 0.392i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.431 - 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.458314 + 0.726909i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.458314 + 0.726909i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (120. - 48.3i)T \) |
good | 2 | \( 1 + (-5.31 + 9.19i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (33.7 - 58.4i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 11 | \( 1 + (261. + 452. i)T + (-8.05e4 + 1.39e5i)T^{2} \) |
| 13 | \( 1 - 76.6T + 3.71e5T^{2} \) |
| 17 | \( 1 + (634. + 1.09e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (-946. + 1.63e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (-575. + 995. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + 3.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + (-5.20e3 - 9.01e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-2.80e3 + 4.85e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 + 1.42e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-5.77e3 + 1.00e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (-2.33e3 - 4.05e3i)T + (-2.09e8 + 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.45e4 + 2.52e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (5.92e3 - 1.02e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-1.81e4 - 3.13e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 1.34e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + (1.30e3 + 2.25e3i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.91e4 + 6.78e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 - 1.67e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (1.83e4 - 3.18e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 - 3.61e4T + 8.58e9T^{2} \) |
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
−13.20449084102411292027066365045, −11.90265694395089440167795258187, −11.14433180065794673534123064835, −10.29083576003623120208454833576, −8.978878853389119796126368033379, −6.70261627687463777516566955716, −5.17712099431459114061882479255, −3.37438777597830535738007355923, −2.77364839949014680663238920139, −0.28655212983406735446375631174,
3.76952127395953585647105708599, 4.79841869661637857084929768211, 6.13349278545672327840794355681, 7.44601866164978588417762108449, 8.313631228617597119834786971848, 9.726903321524565315317125020540, 12.09211254470523033951469051661, 12.90784329519050183164086307811, 13.52795675979752935398348817989, 15.10794116990830141703533933176