L(s) = 1 | + (−2.95 + 9.09i)2-s + (10.1 − 11.8i)3-s + (−48.0 − 34.8i)4-s + (91.6 − 29.7i)5-s + (77.9 + 126. i)6-s + (70.5 − 97.1i)7-s + (211. − 153. i)8-s + (−38.3 − 239. i)9-s + 921. i·10-s + (−277. + 289. i)11-s + (−899. + 216. i)12-s + (390. + 126. i)13-s + (674. + 928. i)14-s + (573. − 1.38e3i)15-s + (185. + 571. i)16-s + (−39.0 − 120. i)17-s + ⋯ |
L(s) = 1 | + (−0.522 + 1.60i)2-s + (0.648 − 0.760i)3-s + (−1.50 − 1.09i)4-s + (1.63 − 0.532i)5-s + (0.883 + 1.44i)6-s + (0.544 − 0.749i)7-s + (1.16 − 0.849i)8-s + (−0.157 − 0.987i)9-s + 2.91i·10-s + (−0.692 + 0.721i)11-s + (−1.80 + 0.434i)12-s + (0.640 + 0.208i)13-s + (0.919 + 1.26i)14-s + (0.658 − 1.59i)15-s + (0.181 + 0.557i)16-s + (−0.0327 − 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.760 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(1.61451 + 0.594987i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61451 + 0.594987i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-10.1 + 11.8i)T \) |
| 11 | \( 1 + (277. - 289. i)T \) |
good | 2 | \( 1 + (2.95 - 9.09i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (-91.6 + 29.7i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-70.5 + 97.1i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-390. - 126. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (39.0 + 120. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.06e3 - 1.46e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 379. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (582. + 423. i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (2.34e3 - 7.22e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (9.26e3 + 6.73e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (6.79e3 - 4.93e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 7.79e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.57e3 - 2.17e3i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (2.56e4 + 8.31e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-7.74e3 + 1.06e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (5.20e3 - 1.69e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 4.77e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-8.27e3 + 2.68e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (8.64e3 - 1.18e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (5.41e4 + 1.75e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-1.54e4 - 4.75e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + 6.99e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (5.48e3 - 1.68e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−15.98236034025590741134931065010, −14.37147993318166517134710775766, −13.93956417127282025751209998394, −12.84108719933912660500926378609, −10.06168084328770647406481238575, −8.971860455645361261354503424719, −7.81052522231118653243384934085, −6.59269317249216279499693810743, −5.27123721749936312930957698969, −1.47423523212476315297335148359,
2.00480776358747034601411836434, 3.08134745786176078446953739087, 5.42466267933240293597294070330, 8.540433964485045148530541998716, 9.459086650309257608557692175163, 10.44621439795323302565909970575, 11.29324495680219363767807995434, 13.22541403566477418541477132686, 13.89355381675160209380143825091, 15.41194362981357422407811918921