| L(s) = 1 | + (−627. − 362. i)2-s + (2.73e4 + 5.23e4i)3-s + (2.62e5 + 4.54e5i)4-s + (−7.64e6 + 4.41e6i)5-s + (1.83e6 − 4.27e7i)6-s + (−7.20e7 + 1.24e8i)7-s − 3.79e8i·8-s + (−1.99e9 + 2.85e9i)9-s + 6.39e9·10-s + (−1.55e10 − 8.95e9i)11-s + (−1.66e10 + 2.61e10i)12-s + (−7.25e10 − 1.25e11i)13-s + (9.03e10 − 5.21e10i)14-s + (−4.39e11 − 2.79e11i)15-s + (−1.37e11 + 2.38e11i)16-s − 2.41e12i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.462 + 0.886i)3-s + (0.249 + 0.433i)4-s + (−0.782 + 0.451i)5-s + (0.0303 − 0.706i)6-s + (−0.255 + 0.441i)7-s − 0.353i·8-s + (−0.572 + 0.820i)9-s + 0.639·10-s + (−0.597 − 0.345i)11-s + (−0.268 + 0.421i)12-s + (−0.526 − 0.911i)13-s + (0.312 − 0.180i)14-s + (−0.762 − 0.485i)15-s + (−0.125 + 0.216i)16-s − 1.19i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.708 + 0.706i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.6966750416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6966750416\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (627. + 362. i)T \) |
| 3 | \( 1 + (-2.73e4 - 5.23e4i)T \) |
| good | 5 | \( 1 + (7.64e6 - 4.41e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (7.20e7 - 1.24e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (1.55e10 + 8.95e9i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (7.25e10 + 1.25e11i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 + 2.41e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 1.78e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (2.36e13 - 1.36e13i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-3.28e14 - 1.89e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (-6.77e14 - 1.17e15i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 + 4.30e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (-1.37e16 + 7.93e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (-1.44e16 + 2.50e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (3.84e16 + 2.21e16i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 + 1.71e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (-1.92e17 + 1.11e17i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (-4.11e17 + 7.12e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-6.59e17 - 1.14e18i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 + 3.46e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 4.16e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (1.16e18 - 2.02e18i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (1.17e19 + 6.78e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 + 7.75e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (4.86e19 - 8.42e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−14.00860684967452426685559131282, −12.18429457473713819538871090157, −10.91298535312441277937835322781, −9.878287302973341610365841128539, −8.553048618207326646294420351043, −7.41866771596508776603775202265, −5.18304211396397738865466454369, −3.42522383451513682602163629133, −2.62118311051351777944455831210, −0.28905379299469920206692849180,
0.863687970988749526157846230196, 2.30816960699843634008116373386, 4.20561908650047523078850100198, 6.30354474733985764203188989850, 7.57591748580409706757309218178, 8.380691467903428318371125326867, 9.874258753243788524912377156673, 11.67067216465727257363208442230, 12.82518150492620653336864613380, 14.23206080926196599049954873415
