L(s) = 1 | + (−12.4 − 21.5i)3-s + (−18.0 − 31.3i)5-s + 208.·7-s + (−187. + 324. i)9-s + 204.·11-s + (−146. + 254. i)13-s + (−449. + 779. i)15-s + (−850. − 1.47e3i)17-s + (−1.32e3 − 841. i)19-s + (−2.59e3 − 4.49e3i)21-s + (1.76e3 − 3.06e3i)23-s + (907. − 1.57e3i)25-s + 3.28e3·27-s + (−3.20e3 + 5.55e3i)29-s + 2.73e3·31-s + ⋯ |
L(s) = 1 | + (−0.797 − 1.38i)3-s + (−0.323 − 0.560i)5-s + 1.61·7-s + (−0.771 + 1.33i)9-s + 0.509·11-s + (−0.240 + 0.417i)13-s + (−0.516 + 0.894i)15-s + (−0.713 − 1.23i)17-s + (−0.844 − 0.535i)19-s + (−1.28 − 2.22i)21-s + (0.696 − 1.20i)23-s + (0.290 − 0.503i)25-s + 0.867·27-s + (−0.707 + 1.22i)29-s + 0.510·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.8258496773\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8258496773\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.32e3 + 841. i)T \) |
good | 3 | \( 1 + (12.4 + 21.5i)T + (-121.5 + 210. i)T^{2} \) |
| 5 | \( 1 + (18.0 + 31.3i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 - 208.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 204.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (146. - 254. i)T + (-1.85e5 - 3.21e5i)T^{2} \) |
| 17 | \( 1 + (850. + 1.47e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 23 | \( 1 + (-1.76e3 + 3.06e3i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (3.20e3 - 5.55e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 - 2.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.42e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (3.69e3 + 6.39e3i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (7.87e3 + 1.36e4i)T + (-7.35e7 + 1.27e8i)T^{2} \) |
| 47 | \( 1 + (9.72e3 - 1.68e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + (4.29e3 - 7.43e3i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + (1.39e4 + 2.41e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (1.53e4 - 2.66e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.41e4 + 4.17e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-3.43e3 - 5.94e3i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-1.93e4 - 3.34e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.21e4 - 5.56e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + 7.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + (2.32e4 - 4.02e4i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + (4.52e4 + 7.82e4i)T + (-4.29e9 + 7.43e9i)T^{2} \) |
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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
−10.84733935920664755263715715873, −8.953409722783035705523713381672, −8.306018163104306662052143694248, −7.20050583209833549375694855883, −6.61404216835313186601908618383, −5.12812162122040948520008577447, −4.57169959648958641142203895814, −2.25504857309452807383032221600, −1.27505483320974880471894553627, −0.25824780312879803766288191070,
1.67117131229118681416416218170, 3.57622435094669537821582452614, 4.44462784559550889963454244796, 5.27964453503448765498198281158, 6.35067883816917002990480577496, 7.75811934958237506308424286838, 8.685346997050333471049570420017, 9.863666586834587792012170157498, 10.73386713068523861900559226315, 11.25969694455143254319936725485