| L(s) = 1 | + (−8.29 − 11.4i)2-s + (4.81 + 14.8i)3-s + (−41.7 + 128. i)4-s + (−55.3 − 40.2i)5-s + (129. − 177. i)6-s + (511. + 166. i)7-s + (954. − 309. i)8-s + (−196. + 142. i)9-s + 965. i·10-s + (287. − 1.29e3i)11-s − 2.10e3·12-s + (−328. − 452. i)13-s + (−2.34e3 − 7.21e3i)14-s + (329. − 1.01e3i)15-s + (−4.45e3 − 3.23e3i)16-s + (453. − 624. i)17-s + ⋯ |
| L(s) = 1 | + (−1.03 − 1.42i)2-s + (0.178 + 0.549i)3-s + (−0.652 + 2.00i)4-s + (−0.442 − 0.321i)5-s + (0.598 − 0.823i)6-s + (1.49 + 0.484i)7-s + (1.86 − 0.605i)8-s + (−0.269 + 0.195i)9-s + 0.965i·10-s + (0.215 − 0.976i)11-s − 1.21·12-s + (−0.149 − 0.205i)13-s + (−0.854 − 2.62i)14-s + (0.0976 − 0.300i)15-s + (−1.08 − 0.790i)16-s + (0.0923 − 0.127i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.804092 - 0.680399i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.804092 - 0.680399i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.81 - 14.8i)T \) |
| 11 | \( 1 + (-287. + 1.29e3i)T \) |
| good | 2 | \( 1 + (8.29 + 11.4i)T + (-19.7 + 60.8i)T^{2} \) |
| 5 | \( 1 + (55.3 + 40.2i)T + (4.82e3 + 1.48e4i)T^{2} \) |
| 7 | \( 1 + (-511. - 166. i)T + (9.51e4 + 6.91e4i)T^{2} \) |
| 13 | \( 1 + (328. + 452. i)T + (-1.49e6 + 4.59e6i)T^{2} \) |
| 17 | \( 1 + (-453. + 624. i)T + (-7.45e6 - 2.29e7i)T^{2} \) |
| 19 | \( 1 + (-5.14e3 + 1.67e3i)T + (3.80e7 - 2.76e7i)T^{2} \) |
| 23 | \( 1 - 1.51e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (-3.67e4 - 1.19e4i)T + (4.81e8 + 3.49e8i)T^{2} \) |
| 31 | \( 1 + (-4.51e4 + 3.27e4i)T + (2.74e8 - 8.44e8i)T^{2} \) |
| 37 | \( 1 + (2.10e4 - 6.48e4i)T + (-2.07e9 - 1.50e9i)T^{2} \) |
| 41 | \( 1 + (9.69e4 - 3.14e4i)T + (3.84e9 - 2.79e9i)T^{2} \) |
| 43 | \( 1 + 2.51e3iT - 6.32e9T^{2} \) |
| 47 | \( 1 + (-2.59e3 - 7.99e3i)T + (-8.72e9 + 6.33e9i)T^{2} \) |
| 53 | \( 1 + (-9.86e4 + 7.16e4i)T + (6.84e9 - 2.10e10i)T^{2} \) |
| 59 | \( 1 + (5.43e4 - 1.67e5i)T + (-3.41e10 - 2.47e10i)T^{2} \) |
| 61 | \( 1 + (-1.56e4 + 2.14e4i)T + (-1.59e10 - 4.89e10i)T^{2} \) |
| 67 | \( 1 - 2.53e5T + 9.04e10T^{2} \) |
| 71 | \( 1 + (5.15e5 + 3.74e5i)T + (3.95e10 + 1.21e11i)T^{2} \) |
| 73 | \( 1 + (3.42e5 + 1.11e5i)T + (1.22e11 + 8.89e10i)T^{2} \) |
| 79 | \( 1 + (-4.08e5 - 5.61e5i)T + (-7.51e10 + 2.31e11i)T^{2} \) |
| 83 | \( 1 + (1.12e5 - 1.55e5i)T + (-1.01e11 - 3.10e11i)T^{2} \) |
| 89 | \( 1 + 6.59e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + (-7.93e5 + 5.76e5i)T + (2.57e11 - 7.92e11i)T^{2} \) |
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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
−15.28621806470336409189190661025, −13.73613957710083076353931811987, −11.91469819649382104102943612474, −11.41692643301948676483560156312, −10.19077583409609775745599748610, −8.712931982754521989927108205869, −8.152828515344527163877089616708, −4.74181411980523403130336752734, −2.93026660622766148838531893252, −1.01704396388412581151863806126,
1.23981587401259750572655448852, 4.95592203705585809121754776729, 6.91649868323063526270835141873, 7.64066784752209816399500426831, 8.703845271146563048279345264032, 10.30245590912911443048572878369, 11.82332780620446237094751815278, 13.96458792487994586490861625546, 14.73459719929526878247926721943, 15.66686259740612989000859570366
