| L(s) = 1 | + (−627. + 362. i)2-s + (−5.61e4 + 1.81e4i)3-s + (2.62e5 − 4.54e5i)4-s + (7.36e6 + 4.25e6i)5-s + (2.86e7 − 3.17e7i)6-s + (−1.38e7 − 2.40e7i)7-s + 3.79e8i·8-s + (2.82e9 − 2.03e9i)9-s − 6.15e9·10-s + (2.91e10 − 1.68e10i)11-s + (−6.49e9 + 3.02e10i)12-s + (2.26e10 − 3.93e10i)13-s + (1.74e10 + 1.00e10i)14-s + (−4.90e11 − 1.05e11i)15-s + (−1.37e11 − 2.38e11i)16-s + 2.72e12i·17-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.951 + 0.307i)3-s + (0.249 − 0.433i)4-s + (0.753 + 0.435i)5-s + (0.474 − 0.524i)6-s + (−0.0491 − 0.0851i)7-s + 0.353i·8-s + (0.811 − 0.584i)9-s − 0.615·10-s + (1.12 − 0.649i)11-s + (−0.104 + 0.488i)12-s + (0.164 − 0.285i)13-s + (0.0601 + 0.0347i)14-s + (−0.851 − 0.182i)15-s + (−0.125 − 0.216i)16-s + 1.35i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (0.716 + 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(0.9228790999\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9228790999\) |
| \(L(11)\) |
|
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 | \( 1 + (5.61e4 - 1.81e4i)T \) |
| good | 5 | \( 1 + (-7.36e6 - 4.25e6i)T + (4.76e13 + 8.25e13i)T^{2} \) |
| 7 | \( 1 + (1.38e7 + 2.40e7i)T + (-3.98e16 + 6.91e16i)T^{2} \) |
| 11 | \( 1 + (-2.91e10 + 1.68e10i)T + (3.36e20 - 5.82e20i)T^{2} \) |
| 13 | \( 1 + (-2.26e10 + 3.93e10i)T + (-9.50e21 - 1.64e22i)T^{2} \) |
| 17 | \( 1 - 2.72e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 + 7.90e12T + 3.75e25T^{2} \) |
| 23 | \( 1 + (5.56e13 + 3.21e13i)T + (8.58e26 + 1.48e27i)T^{2} \) |
| 29 | \( 1 + (-4.02e14 + 2.32e14i)T + (8.84e28 - 1.53e29i)T^{2} \) |
| 31 | \( 1 + (7.34e13 - 1.27e14i)T + (-3.35e29 - 5.81e29i)T^{2} \) |
| 37 | \( 1 + 1.06e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + (4.92e15 + 2.84e15i)T + (9.00e31 + 1.56e32i)T^{2} \) |
| 43 | \( 1 + (-6.49e15 - 1.12e16i)T + (-2.33e32 + 4.04e32i)T^{2} \) |
| 47 | \( 1 + (-6.95e16 + 4.01e16i)T + (1.38e33 - 2.39e33i)T^{2} \) |
| 53 | \( 1 - 2.94e16iT - 3.05e34T^{2} \) |
| 59 | \( 1 + (5.15e17 + 2.97e17i)T + (1.30e35 + 2.26e35i)T^{2} \) |
| 61 | \( 1 + (7.76e16 + 1.34e17i)T + (-2.54e35 + 4.40e35i)T^{2} \) |
| 67 | \( 1 + (-1.25e18 + 2.17e18i)T + (-1.66e36 - 2.87e36i)T^{2} \) |
| 71 | \( 1 + 3.22e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 + 2.20e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + (-4.14e18 - 7.18e18i)T + (-4.48e37 + 7.76e37i)T^{2} \) |
| 83 | \( 1 + (2.53e19 - 1.46e19i)T + (1.20e38 - 2.08e38i)T^{2} \) |
| 89 | \( 1 + 1.64e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + (-2.86e19 - 4.95e19i)T + (-2.71e39 + 4.70e39i)T^{2} \) |
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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
−14.06470887253401657565415546047, −12.29366846338524244229349999312, −10.80156513966136267730359896065, −10.04312126085375405489314777669, −8.518443856135250144732427216612, −6.46182887831714096117301885043, −6.01944344162871299794576229707, −4.07032669563668437700146146025, −1.83335981313715799680389140743, −0.40574064923099379017722120657,
1.07214997582178734182668252585, 2.05239778965719007862749063575, 4.37763503682621893548492532454, 5.97074877498329642959276991865, 7.18728380196220338328878954950, 9.060972762246239746435814190453, 10.12712414727169812085483942839, 11.58049124050846579394002998495, 12.48572024539410378702230385588, 13.86690847877855128812343680269
