L(s) = 1 | + (−13.5 − 7.79i)3-s + (−68.9 + 39.7i)5-s + (−284. + 191. i)7-s + (121.5 + 210. i)9-s + (411. − 712. i)11-s − 2.42e3i·13-s + 1.24e3·15-s + (6.75e3 + 3.89e3i)17-s + (−5.78e3 + 3.34e3i)19-s + (5.33e3 − 376. i)21-s + (9.41e3 + 1.63e4i)23-s + (−4.64e3 + 8.04e3i)25-s − 3.78e3i·27-s + 1.38e4·29-s + (2.41e4 + 1.39e4i)31-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.288i)3-s + (−0.551 + 0.318i)5-s + (−0.828 + 0.559i)7-s + (0.166 + 0.288i)9-s + (0.308 − 0.534i)11-s − 1.10i·13-s + 0.367·15-s + (1.37 + 0.793i)17-s + (−0.843 + 0.487i)19-s + (0.575 − 0.0406i)21-s + (0.773 + 1.34i)23-s + (−0.297 + 0.515i)25-s − 0.192i·27-s + 0.569·29-s + (0.810 + 0.467i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.980 + 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.02555523831\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.02555523831\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (13.5 + 7.79i)T \) |
| 7 | \( 1 + (284. - 191. i)T \) |
good | 5 | \( 1 + (68.9 - 39.7i)T + (7.81e3 - 1.35e4i)T^{2} \) |
| 11 | \( 1 + (-411. + 712. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + 2.42e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-6.75e3 - 3.89e3i)T + (1.20e7 + 2.09e7i)T^{2} \) |
| 19 | \( 1 + (5.78e3 - 3.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-9.41e3 - 1.63e4i)T + (-7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 - 1.38e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-2.41e4 - 1.39e4i)T + (4.43e8 + 7.68e8i)T^{2} \) |
| 37 | \( 1 + (3.98e4 + 6.90e4i)T + (-1.28e9 + 2.22e9i)T^{2} \) |
| 41 | \( 1 + 5.91e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 9.18e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-4.34e3 + 2.50e3i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (9.31e4 - 1.61e5i)T + (-1.10e10 - 1.91e10i)T^{2} \) |
| 59 | \( 1 + (1.95e5 + 1.12e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.25e5 - 7.21e4i)T + (2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.17e5 - 2.03e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 + 9.62e4T + 1.28e11T^{2} \) |
| 73 | \( 1 + (-2.38e5 - 1.37e5i)T + (7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (3.40e5 + 5.90e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + 1.28e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (3.22e5 - 1.86e5i)T + (2.48e11 - 4.30e11i)T^{2} \) |
| 97 | \( 1 + 6.20e5iT - 8.32e11T^{2} \) |
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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
−10.95278255812987882172073370186, −10.31024624033504238886804694053, −9.163392961282550129691554939370, −8.074381741401823512274928032052, −7.24168711220873688917414560353, −6.02640246939663150409744819207, −5.51218325693142719035304664403, −3.77455712049379599132322544173, −2.97750228926324295898233195216, −1.27255198060961256381470753052,
0.007927518594952216247566279742, 1.05896336442767658466613740802, 2.86451577753288930730843048332, 4.18903969489330793096847305243, 4.76958663127782957183811461413, 6.35999379402888048965411263915, 6.93116222007154496803549361610, 8.147843262451880322375634752250, 9.312319636390971076937064346516, 10.00401886024318021916011350319