| L(s) = 1 | + (7.65 − 1.34i)5-s + (10.4 + 8.78i)7-s + (−2.55 − 0.450i)11-s + (−8.23 − 2.99i)13-s + (−15.2 + 8.82i)17-s + (1.46 − 2.54i)19-s + (11.8 + 14.1i)23-s + (33.2 − 12.1i)25-s + (−1.05 − 2.91i)29-s + (30.6 − 25.7i)31-s + (92.0 + 53.1i)35-s + (−12.8 − 22.2i)37-s + (−21.2 + 58.3i)41-s + (5.71 − 32.4i)43-s + (33.4 − 39.8i)47-s + ⋯ |
| L(s) = 1 | + (1.53 − 0.269i)5-s + (1.49 + 1.25i)7-s + (−0.232 − 0.0409i)11-s + (−0.633 − 0.230i)13-s + (−0.899 + 0.519i)17-s + (0.0773 − 0.133i)19-s + (0.517 + 0.616i)23-s + (1.33 − 0.484i)25-s + (−0.0365 − 0.100i)29-s + (0.988 − 0.829i)31-s + (2.62 + 1.51i)35-s + (−0.346 − 0.600i)37-s + (−0.518 + 1.42i)41-s + (0.132 − 0.753i)43-s + (0.711 − 0.847i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.31113 + 0.422651i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.31113 + 0.422651i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-7.65 + 1.34i)T + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-10.4 - 8.78i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (2.55 + 0.450i)T + (113. + 41.3i)T^{2} \) |
| 13 | \( 1 + (8.23 + 2.99i)T + (129. + 108. i)T^{2} \) |
| 17 | \( 1 + (15.2 - 8.82i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-1.46 + 2.54i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-11.8 - 14.1i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (1.05 + 2.91i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-30.6 + 25.7i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (12.8 + 22.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (21.2 - 58.3i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-5.71 + 32.4i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-33.4 + 39.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 + 53.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (102. - 18.0i)T + (3.27e3 - 1.19e3i)T^{2} \) |
| 61 | \( 1 + (4.56 + 3.82i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (44.4 + 16.1i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-77.5 + 44.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-33.6 + 58.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-34.1 + 12.4i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (6.94 + 19.0i)T + (-5.27e3 + 4.42e3i)T^{2} \) |
| 89 | \( 1 + (83.7 + 48.3i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (11.6 - 66.2i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| 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
−11.45062170155645124335884342191, −10.51149609085285924037019332702, −9.443161620778827178039032524606, −8.773367210069100843564411747529, −7.82390488966150499559171809454, −6.30568125895536324953643583831, −5.41446032044318295568096831623, −4.77544030703083975170755537953, −2.50484932441950686638067562446, −1.71744547837376020031051669741,
1.34664389756345237253412837644, 2.53238919728561355477300924974, 4.47952864459783943652509912644, 5.20451066307771418655645654917, 6.58174554968722657721764008746, 7.38171381903568037012325170930, 8.551358460819184520174367553504, 9.625482982235159227575014960015, 10.53981067805587832220419589725, 10.99486621686260693986503880448
