L(s) = 1 | + 5.79e3i·2-s + (−3.88e5 + 1.54e6i)3-s − 3.35e7·4-s + 1.99e8i·5-s + (−8.95e9 − 2.25e9i)6-s − 4.03e9·7-s − 1.94e11i·8-s + (−2.23e12 − 1.20e12i)9-s − 1.15e12·10-s − 4.12e13i·11-s + (1.30e13 − 5.18e13i)12-s − 9.55e13·13-s − 2.33e13i·14-s + (−3.08e14 − 7.74e13i)15-s + 1.12e15·16-s − 5.48e15i·17-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.243 + 0.969i)3-s − 0.500·4-s + 0.163i·5-s + (−0.685 − 0.172i)6-s − 0.0416·7-s − 0.353i·8-s + (−0.881 − 0.472i)9-s − 0.115·10-s − 1.19i·11-s + (0.121 − 0.484i)12-s − 0.315·13-s − 0.0294i·14-s + (−0.158 − 0.0397i)15-s + 0.250·16-s − 0.553i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(27-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+13) \, L(s)\cr =\mathstrut & (0.969 + 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{27}{2})\) |
\(\approx\) |
\(0.8897940130\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8897940130\) |
\(L(14)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 5.79e3iT \) |
| 3 | \( 1 + (3.88e5 - 1.54e6i)T \) |
good | 5 | \( 1 - 1.99e8iT - 1.49e18T^{2} \) |
| 7 | \( 1 + 4.03e9T + 9.38e21T^{2} \) |
| 11 | \( 1 + 4.12e13iT - 1.19e27T^{2} \) |
| 13 | \( 1 + 9.55e13T + 9.17e28T^{2} \) |
| 17 | \( 1 + 5.48e15iT - 9.81e31T^{2} \) |
| 19 | \( 1 + 2.57e16T + 1.76e33T^{2} \) |
| 23 | \( 1 - 6.39e17iT - 2.54e35T^{2} \) |
| 29 | \( 1 + 1.27e19iT - 1.05e38T^{2} \) |
| 31 | \( 1 - 4.74e19T + 5.96e38T^{2} \) |
| 37 | \( 1 + 6.19e19T + 5.93e40T^{2} \) |
| 41 | \( 1 - 1.36e20iT - 8.55e41T^{2} \) |
| 43 | \( 1 + 2.89e21T + 2.95e42T^{2} \) |
| 47 | \( 1 + 1.44e21iT - 2.98e43T^{2} \) |
| 53 | \( 1 + 1.68e22iT - 6.77e44T^{2} \) |
| 59 | \( 1 + 1.69e23iT - 1.10e46T^{2} \) |
| 61 | \( 1 - 1.07e23T + 2.62e46T^{2} \) |
| 67 | \( 1 - 7.95e22T + 3.00e47T^{2} \) |
| 71 | \( 1 + 1.77e24iT - 1.35e48T^{2} \) |
| 73 | \( 1 + 2.59e24T + 2.79e48T^{2} \) |
| 79 | \( 1 - 5.88e24T + 2.17e49T^{2} \) |
| 83 | \( 1 + 7.37e24iT - 7.87e49T^{2} \) |
| 89 | \( 1 + 7.84e24iT - 4.83e50T^{2} \) |
| 97 | \( 1 + 7.65e25T + 4.52e51T^{2} \) |
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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
−16.28828805693883957634362993589, −15.13782598281303317168924074413, −13.73054829117253263735663142511, −11.47767152019588099177340752604, −9.874358103065890881415306037885, −8.386001498549118927465418549093, −6.30888970840638005891640851077, −4.89439105405847051435505365445, −3.26333835483676900168983272453, −0.33471400306106074123376869757,
1.24426973634046856035387547989, 2.55424317825278415833738360212, 4.75697089245883205568186985839, 6.73197568481197597334589843825, 8.469473921179877256023160356645, 10.40523319793172939067129621141, 12.09740905553771183780408249953, 12.96073034760678440321631541977, 14.62587839464113786103846499896, 16.98498161212748471636600994471