| L(s) = 1 | + (−627. + 362. i)2-s + (−2.90e4 − 5.14e4i)3-s + (2.62e5 − 4.54e5i)4-s + (1.35e7 + 7.83e6i)5-s + (3.68e7 + 2.17e7i)6-s + (−1.44e8 − 2.50e8i)7-s + 3.79e8i·8-s + (−1.79e9 + 2.98e9i)9-s − 1.13e10·10-s + (2.16e8 − 1.24e8i)11-s + (−3.09e10 − 2.82e8i)12-s + (−1.02e11 + 1.77e11i)13-s + (1.81e11 + 1.04e11i)14-s + (8.45e9 − 9.24e11i)15-s + (−1.37e11 − 2.38e11i)16-s − 3.27e12i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.492 − 0.870i)3-s + (0.249 − 0.433i)4-s + (1.38 + 0.802i)5-s + (0.609 + 0.359i)6-s + (−0.511 − 0.886i)7-s + 0.353i·8-s + (−0.515 + 0.856i)9-s − 1.13·10-s + (0.00833 − 0.00481i)11-s + (−0.499 − 0.00456i)12-s + (−0.742 + 1.28i)13-s + (0.626 + 0.361i)14-s + (0.0146 − 1.60i)15-s + (−0.125 − 0.216i)16-s − 1.62i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.933 - 0.359i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.09778951556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09778951556\) |
| \(L(11)\) |
|
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 | \( 1 + (2.90e4 + 5.14e4i)T \) |
| good | 5 | \( 1 + (-1.35e7 - 7.83e6i)T + (4.76e13 + 8.25e13i)T^{2} \) |
| 7 | \( 1 + (1.44e8 + 2.50e8i)T + (-3.98e16 + 6.91e16i)T^{2} \) |
| 11 | \( 1 + (-2.16e8 + 1.24e8i)T + (3.36e20 - 5.82e20i)T^{2} \) |
| 13 | \( 1 + (1.02e11 - 1.77e11i)T + (-9.50e21 - 1.64e22i)T^{2} \) |
| 17 | \( 1 + 3.27e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 6.83e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (1.99e13 + 1.15e13i)T + (8.58e26 + 1.48e27i)T^{2} \) |
| 29 | \( 1 + (4.37e13 - 2.52e13i)T + (8.84e28 - 1.53e29i)T^{2} \) |
| 31 | \( 1 + (5.34e13 - 9.26e13i)T + (-3.35e29 - 5.81e29i)T^{2} \) |
| 37 | \( 1 + 6.08e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (1.11e16 + 6.42e15i)T + (9.00e31 + 1.56e32i)T^{2} \) |
| 43 | \( 1 + (1.04e16 + 1.80e16i)T + (-2.33e32 + 4.04e32i)T^{2} \) |
| 47 | \( 1 + (3.71e16 - 2.14e16i)T + (1.38e33 - 2.39e33i)T^{2} \) |
| 53 | \( 1 - 2.87e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (-3.86e17 - 2.23e17i)T + (1.30e35 + 2.26e35i)T^{2} \) |
| 61 | \( 1 + (4.61e17 + 8.00e17i)T + (-2.54e35 + 4.40e35i)T^{2} \) |
| 67 | \( 1 + (1.15e18 - 2.00e18i)T + (-1.66e36 - 2.87e36i)T^{2} \) |
| 71 | \( 1 - 4.08e17iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 5.70e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-7.17e18 - 1.24e19i)T + (-4.48e37 + 7.76e37i)T^{2} \) |
| 83 | \( 1 + (1.76e19 - 1.02e19i)T + (1.20e38 - 2.08e38i)T^{2} \) |
| 89 | \( 1 - 8.44e18iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-9.95e18 - 1.72e19i)T + (-2.71e39 + 4.70e39i)T^{2} \) |
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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
−14.04826783875871621609367292571, −13.75393862403276022869352990561, −11.79912654515877127425672698823, −10.37761431455313301328449855921, −9.407694410281523712697182128258, −7.17092187545982361399171311311, −6.75747417134130044296671194783, −5.33404398934922529573060839229, −2.60353194454592365100350957969, −1.41572203310987583517688430976,
0.03383548484769883386461170509, 1.61887910815663819658786247235, 3.14990853510317735370875735035, 5.20323906582599181071349581906, 6.04160813131255005331607116509, 8.526366963232081011472994546141, 9.665786188165312105633426086937, 10.25703162265022166079681651914, 12.06293435490838515908982306015, 13.06159063716690779128574662357
