L(s) = 1 | + (0.743 − 0.241i)2-s + (−4.20 − 3.05i)3-s + (−12.4 + 9.04i)4-s + (−9.71 + 29.9i)5-s + (−3.86 − 1.25i)6-s + (16.8 + 23.2i)7-s + (−14.4 + 19.8i)8-s + (8.34 + 25.6i)9-s + 24.5i·10-s + (−24.4 − 118. i)11-s + 79.9·12-s + (−294. + 95.8i)13-s + (18.1 + 13.1i)14-s + (132. − 96.0i)15-s + (70.1 − 215. i)16-s + (397. + 129. i)17-s + ⋯ |
L(s) = 1 | + (0.185 − 0.0603i)2-s + (−0.467 − 0.339i)3-s + (−0.778 + 0.565i)4-s + (−0.388 + 1.19i)5-s + (−0.107 − 0.0348i)6-s + (0.344 + 0.474i)7-s + (−0.225 + 0.310i)8-s + (0.103 + 0.317i)9-s + 0.245i·10-s + (−0.202 − 0.979i)11-s + 0.555·12-s + (−1.74 + 0.567i)13-s + (0.0926 + 0.0673i)14-s + (0.587 − 0.426i)15-s + (0.274 − 0.843i)16-s + (1.37 + 0.446i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.514 - 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(0.364545 + 0.643518i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.364545 + 0.643518i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (4.20 + 3.05i)T \) |
| 11 | \( 1 + (24.4 + 118. i)T \) |
good | 2 | \( 1 + (-0.743 + 0.241i)T + (12.9 - 9.40i)T^{2} \) |
| 5 | \( 1 + (9.71 - 29.9i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (-16.8 - 23.2i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (294. - 95.8i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-397. - 129. i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (144. - 198. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 - 529.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (-164. - 226. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (-443. - 1.36e3i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (1.46e3 - 1.06e3i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (289. - 398. i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 287. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (369. + 268. i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (343. + 1.05e3i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (295. - 214. i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-445. - 144. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 - 5.31e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (2.93e3 - 9.04e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (2.51e3 + 3.46e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (6.20e3 - 2.01e3i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-5.63e3 - 1.83e3i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 - 5.36e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-1.24e3 - 3.84e3i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−16.61606938762994911812844079361, −14.81515537860651946043668627160, −14.10665941468955516893307361878, −12.49323202856523311159378319702, −11.66518366176024265381818493102, −10.22171889000362907066465733204, −8.367741932648158524828085131229, −7.07109235744302897027326648432, −5.19556907446692172491711663058, −3.13114127091501939506346088270,
0.55203782018277697319205938013, 4.55455221449560100271389528986, 5.19856206475509559661995302491, 7.60628557719503097262443115163, 9.311500885139189851571434475916, 10.26212234599596019715843134512, 12.11877560492284030750984510762, 12.93667472257351693888355572260, 14.50782141517929288459065847162, 15.44810900208204639935284389138