| L(s) = 1 | + (627. + 362. i)2-s + (−2.37e3 − 5.90e4i)3-s + (2.62e5 + 4.54e5i)4-s + (−6.58e6 + 3.80e6i)5-s + (1.98e7 − 3.78e7i)6-s + (7.46e7 − 1.29e8i)7-s + 3.79e8i·8-s + (−3.47e9 + 2.80e8i)9-s − 5.50e9·10-s + (3.07e10 + 1.77e10i)11-s + (2.61e10 − 1.65e10i)12-s + (−1.12e11 − 1.94e11i)13-s + (9.36e10 − 5.40e10i)14-s + (2.39e11 + 3.79e11i)15-s + (−1.37e11 + 2.38e11i)16-s + 2.48e12i·17-s + ⋯ |
| L(s) = 1 | + (0.612 + 0.353i)2-s + (−0.0402 − 0.999i)3-s + (0.249 + 0.433i)4-s + (−0.674 + 0.389i)5-s + (0.328 − 0.626i)6-s + (0.264 − 0.457i)7-s + 0.353i·8-s + (−0.996 + 0.0804i)9-s − 0.550·10-s + (1.18 + 0.684i)11-s + (0.422 − 0.267i)12-s + (−0.812 − 1.40i)13-s + (0.323 − 0.186i)14-s + (0.415 + 0.657i)15-s + (−0.125 + 0.216i)16-s + 1.23i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0938 - 0.995i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.0938 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(1.766580759\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766580759\) |
| \(L(11)\) |
|
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 | \( 1 + (2.37e3 + 5.90e4i)T \) |
| good | 5 | \( 1 + (6.58e6 - 3.80e6i)T + (4.76e13 - 8.25e13i)T^{2} \) |
| 7 | \( 1 + (-7.46e7 + 1.29e8i)T + (-3.98e16 - 6.91e16i)T^{2} \) |
| 11 | \( 1 + (-3.07e10 - 1.77e10i)T + (3.36e20 + 5.82e20i)T^{2} \) |
| 13 | \( 1 + (1.12e11 + 1.94e11i)T + (-9.50e21 + 1.64e22i)T^{2} \) |
| 17 | \( 1 - 2.48e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 7.60e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (-5.93e12 + 3.42e12i)T + (8.58e26 - 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-6.25e14 - 3.60e14i)T + (8.84e28 + 1.53e29i)T^{2} \) |
| 31 | \( 1 + (-4.26e14 - 7.38e14i)T + (-3.35e29 + 5.81e29i)T^{2} \) |
| 37 | \( 1 - 5.72e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (-3.48e15 + 2.01e15i)T + (9.00e31 - 1.56e32i)T^{2} \) |
| 43 | \( 1 + (1.52e16 - 2.63e16i)T + (-2.33e32 - 4.04e32i)T^{2} \) |
| 47 | \( 1 + (1.80e16 + 1.04e16i)T + (1.38e33 + 2.39e33i)T^{2} \) |
| 53 | \( 1 - 2.64e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (1.59e17 - 9.20e16i)T + (1.30e35 - 2.26e35i)T^{2} \) |
| 61 | \( 1 + (1.11e17 - 1.93e17i)T + (-2.54e35 - 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-2.13e17 - 3.69e17i)T + (-1.66e36 + 2.87e36i)T^{2} \) |
| 71 | \( 1 + 5.24e17iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 4.66e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-1.90e18 + 3.30e18i)T + (-4.48e37 - 7.76e37i)T^{2} \) |
| 83 | \( 1 + (1.63e19 + 9.42e18i)T + (1.20e38 + 2.08e38i)T^{2} \) |
| 89 | \( 1 - 3.96e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-5.68e19 + 9.84e19i)T + (-2.71e39 - 4.70e39i)T^{2} \) |
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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
−14.44909985021466977420155965197, −12.87813900019574522892812342755, −12.09774304618081836478039019504, −10.68535789861776855237948586841, −8.306978710544639550561331553393, −7.28796037484738064303904129547, −6.23802440326976775947856047576, −4.44021685281827449939281313397, −2.94587286745123310971289082067, −1.29591915559201434711118370862,
0.41035781275133480671444247100, 2.42379096631882360821490687101, 4.02807762421584395801887849765, 4.73492080737266186213092467398, 6.39557704121214263487664521973, 8.580182897117419508248301786621, 9.707964279361773271559557393007, 11.53049312226988212520932854585, 11.85638410852672905209726419027, 13.95178921869412321498198496797
