| L(s) = 1 | + (0.402 + 1.23i)2-s + (2.42 + 1.76i)3-s + (5.09 − 3.70i)4-s + (−2.50 + 7.70i)5-s + (−1.20 + 3.71i)6-s + (−0.439 + 0.319i)7-s + (15.0 + 10.9i)8-s + (2.78 + 8.55i)9-s − 10.5·10-s + (−25.3 − 26.2i)11-s + 18.9·12-s + (−21.5 − 66.2i)13-s + (−0.573 − 0.416i)14-s + (−19.6 + 14.2i)15-s + (8.07 − 24.8i)16-s + (−13.1 + 40.4i)17-s + ⋯ |
| L(s) = 1 | + (0.142 + 0.438i)2-s + (0.467 + 0.339i)3-s + (0.637 − 0.463i)4-s + (−0.223 + 0.689i)5-s + (−0.0822 + 0.252i)6-s + (−0.0237 + 0.0172i)7-s + (0.666 + 0.484i)8-s + (0.103 + 0.317i)9-s − 0.333·10-s + (−0.695 − 0.718i)11-s + 0.454·12-s + (−0.459 − 1.41i)13-s + (−0.0109 − 0.00794i)14-s + (−0.338 + 0.245i)15-s + (0.126 − 0.388i)16-s + (−0.187 + 0.577i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.48933 + 0.563034i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48933 + 0.563034i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-2.42 - 1.76i)T \) |
| 11 | \( 1 + (25.3 + 26.2i)T \) |
| good | 2 | \( 1 + (-0.402 - 1.23i)T + (-6.47 + 4.70i)T^{2} \) |
| 5 | \( 1 + (2.50 - 7.70i)T + (-101. - 73.4i)T^{2} \) |
| 7 | \( 1 + (0.439 - 0.319i)T + (105. - 326. i)T^{2} \) |
| 13 | \( 1 + (21.5 + 66.2i)T + (-1.77e3 + 1.29e3i)T^{2} \) |
| 17 | \( 1 + (13.1 - 40.4i)T + (-3.97e3 - 2.88e3i)T^{2} \) |
| 19 | \( 1 + (61.0 + 44.3i)T + (2.11e3 + 6.52e3i)T^{2} \) |
| 23 | \( 1 - 37.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + (97.1 - 70.5i)T + (7.53e3 - 2.31e4i)T^{2} \) |
| 31 | \( 1 + (-85.6 - 263. i)T + (-2.41e4 + 1.75e4i)T^{2} \) |
| 37 | \( 1 + (-12.0 + 8.75i)T + (1.56e4 - 4.81e4i)T^{2} \) |
| 41 | \( 1 + (230. + 167. i)T + (2.12e4 + 6.55e4i)T^{2} \) |
| 43 | \( 1 - 312.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (272. + 198. i)T + (3.20e4 + 9.87e4i)T^{2} \) |
| 53 | \( 1 + (-52.4 - 161. i)T + (-1.20e5 + 8.75e4i)T^{2} \) |
| 59 | \( 1 + (-648. + 471. i)T + (6.34e4 - 1.95e5i)T^{2} \) |
| 61 | \( 1 + (119. - 367. i)T + (-1.83e5 - 1.33e5i)T^{2} \) |
| 67 | \( 1 - 981.T + 3.00e5T^{2} \) |
| 71 | \( 1 + (165. - 509. i)T + (-2.89e5 - 2.10e5i)T^{2} \) |
| 73 | \( 1 + (369. - 268. i)T + (1.20e5 - 3.69e5i)T^{2} \) |
| 79 | \( 1 + (191. + 589. i)T + (-3.98e5 + 2.89e5i)T^{2} \) |
| 83 | \( 1 + (-61.3 + 188. i)T + (-4.62e5 - 3.36e5i)T^{2} \) |
| 89 | \( 1 + 1.31e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + (287. + 884. i)T + (-7.38e5 + 5.36e5i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.93720046069989919498985784586, −15.20306054575639008276736581222, −14.39176188071209403675583505333, −12.94955838108748123748786430960, −10.98699774420376353527877568759, −10.35819289744519812481625619101, −8.323657405723401780741516538951, −7.00013058190785902411759176057, −5.37100274233153274389910985992, −2.91306726018415625311837657410,
2.21276575554957768119308440266, 4.34674395697612340356973556155, 6.89319904065262219410525388300, 8.134466055107304511767186425890, 9.708635260920208721876541130115, 11.43189476437249042375013677990, 12.44157359150186523180099856818, 13.33632301986327462823426910699, 14.90185990453697554641682239040, 16.17812289075082461329091368192
