| L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.344 − 1.69i)3-s + (0.766 + 0.642i)4-s + (−0.445 + 2.52i)5-s + (−0.904 + 1.47i)6-s + (−0.766 + 0.642i)7-s + (−0.500 − 0.866i)8-s + (−2.76 − 1.17i)9-s + (1.28 − 2.22i)10-s + (1.12 + 6.39i)11-s + (1.35 − 1.07i)12-s + (−5.08 + 1.84i)13-s + (0.939 − 0.342i)14-s + (4.13 + 1.62i)15-s + (0.173 + 0.984i)16-s + (−1.82 + 3.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.664 − 0.241i)2-s + (0.199 − 0.979i)3-s + (0.383 + 0.321i)4-s + (−0.199 + 1.12i)5-s + (−0.369 + 0.602i)6-s + (−0.289 + 0.242i)7-s + (−0.176 − 0.306i)8-s + (−0.920 − 0.390i)9-s + (0.405 − 0.702i)10-s + (0.339 + 1.92i)11-s + (0.391 − 0.311i)12-s + (−1.40 + 0.513i)13-s + (0.251 − 0.0914i)14-s + (1.06 + 0.420i)15-s + (0.0434 + 0.246i)16-s + (−0.442 + 0.766i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.193 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.488504 + 0.401770i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.488504 + 0.401770i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.939 + 0.342i)T \) |
| 3 | \( 1 + (-0.344 + 1.69i)T \) |
| 7 | \( 1 + (0.766 - 0.642i)T \) |
| good | 5 | \( 1 + (0.445 - 2.52i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.12 - 6.39i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (5.08 - 1.84i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.82 - 3.15i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.24 + 5.62i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.00 - 0.840i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (1.09 + 0.398i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-6.17 - 5.18i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-1.49 + 2.58i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-8.76 + 3.19i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.201 - 1.14i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (2.71 - 2.28i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 2.63T + 53T^{2} \) |
| 59 | \( 1 + (1.04 - 5.90i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.25 - 1.89i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (-8.09 + 2.94i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (3.79 - 6.57i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.44 + 11.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.00 - 2.54i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (11.0 + 4.02i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (0.396 + 0.686i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1.29 + 7.33i)T + (-91.1 + 33.1i)T^{2} \) |
| 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
−11.59534278319618618616789346027, −10.64755094432151317894007868823, −9.673804479934801645374880668623, −8.891100295294878618772108277839, −7.53303541757973956056252216746, −7.04066185732638567297215904970, −6.43491206168099399401008972769, −4.50331449962707898061155965006, −2.76590158137809375437633088513, −2.05480878092854150807734862835,
0.48558269928795126425010524512, 2.85528890045466209444957462256, 4.22555664836114817357498982890, 5.28887039604113394188165542119, 6.24296048586744533430576797329, 7.902386251845116060966568706201, 8.460651221342281270093106927896, 9.308687714823408814651203177510, 10.00809450639942320235702794912, 10.99897395734911477418849563800
