L(s) = 1 | + (79.9 + 42.3i)2-s + (805. − 805. i)3-s + (4.60e3 + 6.77e3i)4-s + (1.88e4 + 1.88e4i)5-s + (9.85e4 − 3.03e4i)6-s − 1.73e4i·7-s + (8.18e4 + 7.36e5i)8-s + 2.95e5i·9-s + (7.08e5 + 2.30e6i)10-s + (8.32e5 + 8.32e5i)11-s + (9.17e6 + 1.74e6i)12-s + (1.44e7 − 1.44e7i)13-s + (7.33e5 − 1.38e6i)14-s + 3.03e7·15-s + (−2.46e7 + 6.24e7i)16-s + 6.64e7·17-s + ⋯ |
L(s) = 1 | + (0.883 + 0.467i)2-s + (0.638 − 0.638i)3-s + (0.562 + 0.826i)4-s + (0.538 + 0.538i)5-s + (0.862 − 0.265i)6-s − 0.0556i·7-s + (0.110 + 0.993i)8-s + 0.185i·9-s + (0.224 + 0.728i)10-s + (0.141 + 0.141i)11-s + (0.886 + 0.168i)12-s + (0.831 − 0.831i)13-s + (0.0260 − 0.0492i)14-s + 0.687·15-s + (−0.367 + 0.930i)16-s + 0.667·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(7)\) |
\(\approx\) |
\(3.95337 + 1.67621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.95337 + 1.67621i\) |
\(L(\frac{15}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-79.9 - 42.3i)T \) |
good | 3 | \( 1 + (-805. + 805. i)T - 1.59e6iT^{2} \) |
| 5 | \( 1 + (-1.88e4 - 1.88e4i)T + 1.22e9iT^{2} \) |
| 7 | \( 1 + 1.73e4iT - 9.68e10T^{2} \) |
| 11 | \( 1 + (-8.32e5 - 8.32e5i)T + 3.45e13iT^{2} \) |
| 13 | \( 1 + (-1.44e7 + 1.44e7i)T - 3.02e14iT^{2} \) |
| 17 | \( 1 - 6.64e7T + 9.90e15T^{2} \) |
| 19 | \( 1 + (6.64e7 - 6.64e7i)T - 4.20e16iT^{2} \) |
| 23 | \( 1 - 3.89e8iT - 5.04e17T^{2} \) |
| 29 | \( 1 + (6.97e8 - 6.97e8i)T - 1.02e19iT^{2} \) |
| 31 | \( 1 - 1.62e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + (2.11e10 + 2.11e10i)T + 2.43e20iT^{2} \) |
| 41 | \( 1 + 4.22e10iT - 9.25e20T^{2} \) |
| 43 | \( 1 + (3.37e10 + 3.37e10i)T + 1.71e21iT^{2} \) |
| 47 | \( 1 + 6.25e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + (1.16e10 + 1.16e10i)T + 2.60e22iT^{2} \) |
| 59 | \( 1 + (3.74e11 + 3.74e11i)T + 1.04e23iT^{2} \) |
| 61 | \( 1 + (1.50e10 - 1.50e10i)T - 1.61e23iT^{2} \) |
| 67 | \( 1 + (-3.31e11 + 3.31e11i)T - 5.48e23iT^{2} \) |
| 71 | \( 1 - 1.41e12iT - 1.16e24T^{2} \) |
| 73 | \( 1 + 3.43e11iT - 1.67e24T^{2} \) |
| 79 | \( 1 + 2.30e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + (2.56e12 - 2.56e12i)T - 8.87e24iT^{2} \) |
| 89 | \( 1 - 3.79e12iT - 2.19e25T^{2} \) |
| 97 | \( 1 - 9.19e12T + 6.73e25T^{2} \) |
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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
−15.75921998275866267789673825460, −14.38010174882765779656998360388, −13.63013897276770948979421867489, −12.47207866986326649162547089260, −10.64629685084155257909623854485, −8.349846217335965634807825599122, −7.08332563972949674588387811691, −5.61104579244252221639803801597, −3.40600804941579608052018171080, −1.96917510172825294100985517470,
1.40938890160941231365386514279, 3.22255912866698159047281227846, 4.61215871110015316353600766747, 6.29236407171435000070656500844, 8.888458169815744112981996343493, 10.06907388177970434055210984581, 11.71338441024596960613702788924, 13.20402002538056009651132766470, 14.26863883046432546663167965463, 15.40032432037402402644655184095