| L(s) = 1 | + (6.48 − 2.10i)2-s + (−4.20 − 3.05i)3-s + (24.6 − 17.9i)4-s + (2.10 − 6.48i)5-s + (−33.6 − 10.9i)6-s + (12.4 + 17.1i)7-s + (58.0 − 79.8i)8-s + (8.34 + 25.6i)9-s − 46.5i·10-s + (32.4 + 116. i)11-s − 158.·12-s + (−221. + 71.9i)13-s + (116. + 84.8i)14-s + (−28.6 + 20.8i)15-s + (57.2 − 176. i)16-s + (146. + 47.4i)17-s + ⋯ |
| L(s) = 1 | + (1.62 − 0.526i)2-s + (−0.467 − 0.339i)3-s + (1.54 − 1.11i)4-s + (0.0843 − 0.259i)5-s + (−0.935 − 0.304i)6-s + (0.253 + 0.349i)7-s + (0.906 − 1.24i)8-s + (0.103 + 0.317i)9-s − 0.465i·10-s + (0.268 + 0.963i)11-s − 1.09·12-s + (−1.31 + 0.425i)13-s + (0.595 + 0.432i)14-s + (−0.127 + 0.0925i)15-s + (0.223 − 0.687i)16-s + (0.505 + 0.164i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.570 + 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.47152 - 1.29179i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.47152 - 1.29179i\) |
| \(L(3)\) |
|
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 | \( 1 + (-32.4 - 116. i)T \) |
| good | 2 | \( 1 + (-6.48 + 2.10i)T + (12.9 - 9.40i)T^{2} \) |
| 5 | \( 1 + (-2.10 + 6.48i)T + (-505. - 367. i)T^{2} \) |
| 7 | \( 1 + (-12.4 - 17.1i)T + (-741. + 2.28e3i)T^{2} \) |
| 13 | \( 1 + (221. - 71.9i)T + (2.31e4 - 1.67e4i)T^{2} \) |
| 17 | \( 1 + (-146. - 47.4i)T + (6.75e4 + 4.90e4i)T^{2} \) |
| 19 | \( 1 + (-308. + 424. i)T + (-4.02e4 - 1.23e5i)T^{2} \) |
| 23 | \( 1 + 911.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (285. + 393. i)T + (-2.18e5 + 6.72e5i)T^{2} \) |
| 31 | \( 1 + (414. + 1.27e3i)T + (-7.47e5 + 5.42e5i)T^{2} \) |
| 37 | \( 1 + (-1.16e3 + 845. i)T + (5.79e5 - 1.78e6i)T^{2} \) |
| 41 | \( 1 + (257. - 354. i)T + (-8.73e5 - 2.68e6i)T^{2} \) |
| 43 | \( 1 + 293. iT - 3.41e6T^{2} \) |
| 47 | \( 1 + (2.19e3 + 1.59e3i)T + (1.50e6 + 4.64e6i)T^{2} \) |
| 53 | \( 1 + (-1.30e3 - 4.01e3i)T + (-6.38e6 + 4.63e6i)T^{2} \) |
| 59 | \( 1 + (983. - 714. i)T + (3.74e6 - 1.15e7i)T^{2} \) |
| 61 | \( 1 + (-1.66e3 - 541. i)T + (1.12e7 + 8.13e6i)T^{2} \) |
| 67 | \( 1 + 1.52e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + (1.68e3 - 5.18e3i)T + (-2.05e7 - 1.49e7i)T^{2} \) |
| 73 | \( 1 + (-1.89e3 - 2.60e3i)T + (-8.77e6 + 2.70e7i)T^{2} \) |
| 79 | \( 1 + (1.36e3 - 442. i)T + (3.15e7 - 2.28e7i)T^{2} \) |
| 83 | \( 1 + (-1.15e3 - 374. i)T + (3.83e7 + 2.78e7i)T^{2} \) |
| 89 | \( 1 + 4.62e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + (-3.52e3 - 1.08e4i)T + (-7.16e7 + 5.20e7i)T^{2} \) |
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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
−15.28923165383967539978874243971, −14.39163027454035819362975168809, −13.15041391753987053609377619017, −12.16282146931739670051270179386, −11.50750734225763691322953228900, −9.777407096807399974379755986519, −7.23309886551146857042010787163, −5.61630649045056150617105739699, −4.46354267820210077983067610398, −2.18405573154702847696151896028,
3.40628573941044758045783206225, 4.99789894406956131086964569529, 6.17015961815497382290815254510, 7.67097015950445081734316064092, 10.14958597341950356262318502448, 11.65719408519625600645192650465, 12.55042303711897867187692410653, 14.13120439785323228827584062666, 14.54811785482386111534665379920, 16.06203388632854634434949091079
