| L(s) = 1 | + (−1.47 + 0.911i)3-s + (−0.412 + 2.87i)5-s + (−0.273 − 2.86i)7-s + (1.33 − 2.68i)9-s + (4.55 − 1.82i)11-s + (1.25 + 1.32i)13-s + (−2.00 − 4.60i)15-s + (−1.49 + 0.517i)17-s + (3.17 + 0.303i)19-s + (3.01 + 3.97i)21-s + (5.20 − 0.248i)23-s + (−3.27 − 0.960i)25-s + (0.480 + 5.17i)27-s + (−0.761 + 0.439i)29-s + (−2.73 + 2.86i)31-s + ⋯ |
| L(s) = 1 | + (−0.850 + 0.526i)3-s + (−0.184 + 1.28i)5-s + (−0.103 − 1.08i)7-s + (0.445 − 0.895i)9-s + (1.37 − 0.549i)11-s + (0.349 + 0.366i)13-s + (−0.518 − 1.18i)15-s + (−0.362 + 0.125i)17-s + (0.728 + 0.0696i)19-s + (0.659 + 0.867i)21-s + (1.08 − 0.0517i)23-s + (−0.654 − 0.192i)25-s + (0.0924 + 0.995i)27-s + (−0.141 + 0.0816i)29-s + (−0.491 + 0.515i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.05190 + 0.575094i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05190 + 0.575094i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.47 - 0.911i)T \) |
| 67 | \( 1 + (-3.34 + 7.46i)T \) |
| good | 5 | \( 1 + (0.412 - 2.87i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (0.273 + 2.86i)T + (-6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (-4.55 + 1.82i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (-1.25 - 1.32i)T + (-0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (1.49 - 0.517i)T + (13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (-3.17 - 0.303i)T + (18.6 + 3.59i)T^{2} \) |
| 23 | \( 1 + (-5.20 + 0.248i)T + (22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (0.761 - 0.439i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.73 - 2.86i)T + (-1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (2.86 - 4.96i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (6.58 + 1.26i)T + (38.0 + 15.2i)T^{2} \) |
| 43 | \( 1 + (-6.45 - 5.59i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (4.64 - 9.01i)T + (-27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (-4.94 - 5.70i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.523 + 1.78i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-4.12 + 10.3i)T + (-44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (-4.94 - 1.71i)T + (55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (-6.17 - 2.47i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (-11.3 + 2.75i)T + (70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (0.0980 - 0.124i)T + (-19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (1.62 - 2.52i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (-6.23 - 3.60i)T + (48.5 + 84.0i)T^{2} \) |
| 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
−10.64916388035475913659841067656, −9.723283671521312673650399456238, −8.950104723246415596274749023452, −7.50035411052734729907673833526, −6.65049118002168759304608594089, −6.39605949168583745535206356445, −4.96506141784140445071374829795, −3.83461783943254818061970433758, −3.31948804569963675880615811140, −1.13229187376344770061376252202,
0.876514044707872246186761543245, 2.04821371594527754952426892575, 3.87403840035776165364120784403, 5.04689889283236266834969450629, 5.52953021279486798985978294098, 6.59412954709613257882013014116, 7.43428751611173166186101681102, 8.696126962109829815695451681532, 9.017644601800259948907902801529, 10.05637187000573440404150941464
