L(s) = 1 | + (−36.4 − 26.8i)2-s + (−475. + 475. i)3-s + (610. + 1.95e3i)4-s + (2.80e3 + 2.80e3i)5-s + (3.01e4 − 4.59e3i)6-s + 7.60e4i·7-s + (3.01e4 − 8.76e4i)8-s − 2.75e5i·9-s + (−2.70e4 − 1.77e5i)10-s + (4.09e5 + 4.09e5i)11-s + (−1.22e6 − 6.39e5i)12-s + (−1.80e6 + 1.80e6i)13-s + (2.03e6 − 2.77e6i)14-s − 2.67e6·15-s + (−3.44e6 + 2.38e6i)16-s − 1.50e6·17-s + ⋯ |
L(s) = 1 | + (−0.805 − 0.592i)2-s + (−1.13 + 1.13i)3-s + (0.298 + 0.954i)4-s + (0.401 + 0.401i)5-s + (1.58 − 0.241i)6-s + 1.70i·7-s + (0.325 − 0.945i)8-s − 1.55i·9-s + (−0.0856 − 0.561i)10-s + (0.767 + 0.767i)11-s + (−1.41 − 0.742i)12-s + (−1.34 + 1.34i)13-s + (1.01 − 1.37i)14-s − 0.907·15-s + (−0.822 + 0.569i)16-s − 0.257·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.977 + 0.211i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(6)\) |
\(\approx\) |
\(0.0574120 - 0.537442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0574120 - 0.537442i\) |
\(L(\frac{13}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (36.4 + 26.8i)T \) |
good | 3 | \( 1 + (475. - 475. i)T - 1.77e5iT^{2} \) |
| 5 | \( 1 + (-2.80e3 - 2.80e3i)T + 4.88e7iT^{2} \) |
| 7 | \( 1 - 7.60e4iT - 1.97e9T^{2} \) |
| 11 | \( 1 + (-4.09e5 - 4.09e5i)T + 2.85e11iT^{2} \) |
| 13 | \( 1 + (1.80e6 - 1.80e6i)T - 1.79e12iT^{2} \) |
| 17 | \( 1 + 1.50e6T + 3.42e13T^{2} \) |
| 19 | \( 1 + (-5.45e6 + 5.45e6i)T - 1.16e14iT^{2} \) |
| 23 | \( 1 - 2.28e6iT - 9.52e14T^{2} \) |
| 29 | \( 1 + (-4.38e7 + 4.38e7i)T - 1.22e16iT^{2} \) |
| 31 | \( 1 - 2.76e7T + 2.54e16T^{2} \) |
| 37 | \( 1 + (-2.40e8 - 2.40e8i)T + 1.77e17iT^{2} \) |
| 41 | \( 1 + 1.08e8iT - 5.50e17T^{2} \) |
| 43 | \( 1 + (-1.91e8 - 1.91e8i)T + 9.29e17iT^{2} \) |
| 47 | \( 1 + 1.72e9T + 2.47e18T^{2} \) |
| 53 | \( 1 + (8.00e7 + 8.00e7i)T + 9.26e18iT^{2} \) |
| 59 | \( 1 + (-6.70e9 - 6.70e9i)T + 3.01e19iT^{2} \) |
| 61 | \( 1 + (-9.82e7 + 9.82e7i)T - 4.35e19iT^{2} \) |
| 67 | \( 1 + (-6.92e9 + 6.92e9i)T - 1.22e20iT^{2} \) |
| 71 | \( 1 - 1.20e9iT - 2.31e20T^{2} \) |
| 73 | \( 1 + 2.00e10iT - 3.13e20T^{2} \) |
| 79 | \( 1 + 1.65e10T + 7.47e20T^{2} \) |
| 83 | \( 1 + (-2.19e10 + 2.19e10i)T - 1.28e21iT^{2} \) |
| 89 | \( 1 - 3.11e10iT - 2.77e21T^{2} \) |
| 97 | \( 1 + 1.12e9T + 7.15e21T^{2} \) |
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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
−17.26607966975362351372110868625, −16.11389754294270708455843553023, −14.90724820492393226511100224341, −12.08642388610681648247074340590, −11.59031980970150623287850995325, −9.897267610867766652672962081997, −9.220824148973190124011392546589, −6.51663926697468375234917350892, −4.65965740607356029208393653174, −2.31480598586036920486554964640,
0.40195874175706597785959418791, 1.20262085526861008207412444628, 5.35879920151058858872709998089, 6.77473723973890520267962704141, 7.78309675236251760812770938912, 10.03027555040143812085519600626, 11.27000732313250356815578449649, 12.96061883975486308706455032506, 14.24207702201509748812870975468, 16.49472197970787008278510620163