| L(s) = 1 | + (0.330 − 1.97i)2-s + (−1.74 − 2.44i)3-s + (−3.78 − 1.30i)4-s + (1.10 + 6.29i)5-s + (−5.39 + 2.63i)6-s + (7.68 + 6.45i)7-s + (−3.82 + 7.02i)8-s + (−2.90 + 8.51i)9-s + (12.7 − 0.108i)10-s + (−0.377 + 2.14i)11-s + (3.41 + 11.5i)12-s + (−2.57 + 7.07i)13-s + (15.2 − 13.0i)14-s + (13.4 − 13.6i)15-s + (12.5 + 9.86i)16-s + (−4.39 + 2.53i)17-s + ⋯ |
| L(s) = 1 | + (0.165 − 0.986i)2-s + (−0.581 − 0.813i)3-s + (−0.945 − 0.326i)4-s + (0.221 + 1.25i)5-s + (−0.898 + 0.439i)6-s + (1.09 + 0.921i)7-s + (−0.477 + 0.878i)8-s + (−0.323 + 0.946i)9-s + (1.27 − 0.0108i)10-s + (−0.0343 + 0.194i)11-s + (0.284 + 0.958i)12-s + (−0.198 + 0.544i)13-s + (1.09 − 0.930i)14-s + (0.894 − 0.912i)15-s + (0.787 + 0.616i)16-s + (−0.258 + 0.149i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0841i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.17566 + 0.0495278i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.17566 + 0.0495278i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.330 + 1.97i)T \) |
| 3 | \( 1 + (1.74 + 2.44i)T \) |
| good | 5 | \( 1 + (-1.10 - 6.29i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-7.68 - 6.45i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (0.377 - 2.14i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (2.57 - 7.07i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (4.39 - 2.53i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (23.2 + 13.4i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-24.4 - 29.1i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (22.1 - 8.05i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (-29.5 + 24.8i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (30.9 - 17.8i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (6.80 - 18.6i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-57.5 - 10.1i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (27.3 - 32.6i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 5.36T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-16.1 - 91.7i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-40.8 + 48.6i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-15.8 + 43.6i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-27.3 + 15.8i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-46.0 + 79.7i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (134. - 48.7i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-44.1 + 16.0i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (44.3 + 25.5i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (11.2 - 63.8i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| 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
−11.84112085602878301840944332889, −11.22224690916540776760058170337, −10.70766658050851051842512453693, −9.305506603478119377950590490609, −8.166617959572343186221173044206, −6.90638567467544989498996431876, −5.77125025428699779774114294298, −4.67145497037850819708953104496, −2.69930935726928117473149345098, −1.79697757303182760854090953120,
0.69981534736221986978091498175, 4.04591594294596260712250323827, 4.79326211029539601173209200766, 5.52365543411637434501846724301, 6.87031566180833407716409838102, 8.290934349346094592848136997917, 8.809517819205471015315468190585, 10.08468606330941284922549421723, 10.95368088103532592901778636569, 12.32541789973693079702194135061
