| L(s) = 1 | + (1.29 + 0.943i)2-s + (−0.141 + 5.19i)3-s + (−1.67 − 5.15i)4-s + (10.7 + 14.7i)5-s + (−5.08 + 6.61i)6-s + (−4.11 + 1.33i)7-s + (6.65 − 20.4i)8-s + (−26.9 − 1.47i)9-s + 29.2i·10-s + (12.8 − 34.1i)11-s + (27.0 − 7.97i)12-s + (37.8 − 52.0i)13-s + (−6.60 − 2.14i)14-s + (−78.2 + 53.6i)15-s + (−7.13 + 5.18i)16-s + (−15.5 + 11.2i)17-s + ⋯ |
| L(s) = 1 | + (0.459 + 0.333i)2-s + (−0.0272 + 0.999i)3-s + (−0.209 − 0.644i)4-s + (0.959 + 1.32i)5-s + (−0.345 + 0.449i)6-s + (−0.222 + 0.0721i)7-s + (0.294 − 0.905i)8-s + (−0.998 − 0.0545i)9-s + 0.926i·10-s + (0.352 − 0.935i)11-s + (0.650 − 0.191i)12-s + (0.807 − 1.11i)13-s + (−0.126 − 0.0409i)14-s + (−1.34 + 0.923i)15-s + (−0.111 + 0.0810i)16-s + (−0.221 + 0.160i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.440 - 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.33347 + 0.830591i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33347 + 0.830591i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.141 - 5.19i)T \) |
| 11 | \( 1 + (-12.8 + 34.1i)T \) |
| good | 2 | \( 1 + (-1.29 - 0.943i)T + (2.47 + 7.60i)T^{2} \) |
| 5 | \( 1 + (-10.7 - 14.7i)T + (-38.6 + 118. i)T^{2} \) |
| 7 | \( 1 + (4.11 - 1.33i)T + (277. - 201. i)T^{2} \) |
| 13 | \( 1 + (-37.8 + 52.0i)T + (-678. - 2.08e3i)T^{2} \) |
| 17 | \( 1 + (15.5 - 11.2i)T + (1.51e3 - 4.67e3i)T^{2} \) |
| 19 | \( 1 + (53.1 + 17.2i)T + (5.54e3 + 4.03e3i)T^{2} \) |
| 23 | \( 1 - 47.1iT - 1.21e4T^{2} \) |
| 29 | \( 1 + (-86.3 - 265. i)T + (-1.97e4 + 1.43e4i)T^{2} \) |
| 31 | \( 1 + (64.0 + 46.5i)T + (9.20e3 + 2.83e4i)T^{2} \) |
| 37 | \( 1 + (3.49 + 10.7i)T + (-4.09e4 + 2.97e4i)T^{2} \) |
| 41 | \( 1 + (-38.9 + 120. i)T + (-5.57e4 - 4.05e4i)T^{2} \) |
| 43 | \( 1 + 324. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + (175. + 57.0i)T + (8.39e4 + 6.10e4i)T^{2} \) |
| 53 | \( 1 + (16.3 - 22.5i)T + (-4.60e4 - 1.41e5i)T^{2} \) |
| 59 | \( 1 + (262. - 85.4i)T + (1.66e5 - 1.20e5i)T^{2} \) |
| 61 | \( 1 + (-278. - 382. i)T + (-7.01e4 + 2.15e5i)T^{2} \) |
| 67 | \( 1 + 649.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (-89.6 - 123. i)T + (-1.10e5 + 3.40e5i)T^{2} \) |
| 73 | \( 1 + (-469. + 152. i)T + (3.14e5 - 2.28e5i)T^{2} \) |
| 79 | \( 1 + (-26.6 + 36.6i)T + (-1.52e5 - 4.68e5i)T^{2} \) |
| 83 | \( 1 + (-352. + 255. i)T + (1.76e5 - 5.43e5i)T^{2} \) |
| 89 | \( 1 + 70.8iT - 7.04e5T^{2} \) |
| 97 | \( 1 + (-970. - 704. i)T + (2.82e5 + 8.68e5i)T^{2} \) |
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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
−16.07629161387745505083470439959, −15.06665245742152889343507024058, −14.28191369055258225873502470814, −13.36253827452013480447075434043, −10.86696205625356642879767814260, −10.38910421266801707092087539789, −9.028132567783909385057010213610, −6.42734862911907647181062450115, −5.54656202975042461750539650085, −3.38706173411862059065640609729,
1.90370789866851791809603299356, 4.57195577254696377599354022078, 6.37592855218490818606791184063, 8.244932202753438305834780602716, 9.334300306476073805329658747255, 11.62528073430540374049656111799, 12.63975165445071138802113492521, 13.27641964697950948695160657589, 14.18597784810506218569950273358, 16.48146810183384475026862984143
