L(s) = 1 | + (−30.9 + 33.0i)2-s + (585. + 585. i)3-s + (−133. − 2.04e3i)4-s + (−6.85e3 + 6.85e3i)5-s + (−3.74e4 + 1.22e3i)6-s − 1.99e4i·7-s + (7.16e4 + 5.88e4i)8-s + 5.08e5i·9-s + (−1.43e4 − 4.38e5i)10-s + (−9.64e4 + 9.64e4i)11-s + (1.11e6 − 1.27e6i)12-s + (−6.74e5 − 6.74e5i)13-s + (6.57e5 + 6.16e5i)14-s − 8.03e6·15-s + (−4.15e6 + 5.46e5i)16-s + 3.70e6·17-s + ⋯ |
L(s) = 1 | + (−0.683 + 0.729i)2-s + (1.39 + 1.39i)3-s + (−0.0652 − 0.997i)4-s + (−0.981 + 0.981i)5-s + (−1.96 + 0.0642i)6-s − 0.447i·7-s + (0.772 + 0.634i)8-s + 2.87i·9-s + (−0.0453 − 1.38i)10-s + (−0.180 + 0.180i)11-s + (1.29 − 1.47i)12-s + (−0.503 − 0.503i)13-s + (0.326 + 0.306i)14-s − 2.73·15-s + (−0.991 + 0.130i)16-s + 0.633·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.866 + 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.323161 - 1.20673i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.323161 - 1.20673i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (30.9 - 33.0i)T \) |
good | 3 | \( 1 + (-585. - 585. i)T + 1.77e5iT^{2} \) |
| 5 | \( 1 + (6.85e3 - 6.85e3i)T - 4.88e7iT^{2} \) |
| 7 | \( 1 + 1.99e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (9.64e4 - 9.64e4i)T - 2.85e11iT^{2} \) |
| 13 | \( 1 + (6.74e5 + 6.74e5i)T + 1.79e12iT^{2} \) |
| 17 | \( 1 - 3.70e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (2.61e6 + 2.61e6i)T + 1.16e14iT^{2} \) |
| 23 | \( 1 + 2.97e7iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-4.14e7 - 4.14e7i)T + 1.22e16iT^{2} \) |
| 31 | \( 1 - 6.07e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (4.37e8 - 4.37e8i)T - 1.77e17iT^{2} \) |
| 41 | \( 1 - 1.20e9iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (1.45e8 - 1.45e8i)T - 9.29e17iT^{2} \) |
| 47 | \( 1 - 1.52e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (2.52e9 - 2.52e9i)T - 9.26e18iT^{2} \) |
| 59 | \( 1 + (4.54e9 - 4.54e9i)T - 3.01e19iT^{2} \) |
| 61 | \( 1 + (-9.96e7 - 9.96e7i)T + 4.35e19iT^{2} \) |
| 67 | \( 1 + (-7.32e9 - 7.32e9i)T + 1.22e20iT^{2} \) |
| 71 | \( 1 - 1.31e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 2.59e9iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 4.08e9T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-3.03e10 - 3.03e10i)T + 1.28e21iT^{2} \) |
| 89 | \( 1 + 5.68e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 5.73e10T + 7.15e21T^{2} \) |
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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
−16.73717181730721466730860877238, −15.54622489915309019908774449605, −14.93723681412329129132574511194, −14.01507916185653770540552906523, −10.76654396898039693282678046114, −9.989480941281818736622296361774, −8.400972992053731004542324916463, −7.40308974551975897691961843387, −4.56142206897114557916900140381, −2.93811567187988018539479471950,
0.57503939181848003014329362622, 2.00081415741020161156402304292, 3.61665506502984511789911828194, 7.38932696691650370609757519566, 8.325278434336474833027695254909, 9.241823303791776572141905188733, 12.00941006338072447720172137799, 12.50063373807227503374166771646, 13.87348316300619107536043130641, 15.64199057645156149395812799438