L(s) = 1 | + (0.156 + 0.0904i)2-s + (−0.983 − 1.70i)4-s + (−2.32 + 1.34i)5-s + (−0.393 − 2.61i)7-s − 0.717i·8-s − 0.485·10-s + 2.69i·11-s + (1.92 + 3.05i)13-s + (0.174 − 0.445i)14-s + (−1.90 + 3.29i)16-s + (−2.38 − 4.12i)17-s + 0.188i·19-s + (4.57 + 2.64i)20-s + (−0.243 + 0.421i)22-s + (−2.19 + 3.80i)23-s + ⋯ |
L(s) = 1 | + (0.110 + 0.0639i)2-s + (−0.491 − 0.851i)4-s + (−1.04 + 0.600i)5-s + (−0.148 − 0.988i)7-s − 0.253i·8-s − 0.153·10-s + 0.812i·11-s + (0.532 + 0.846i)13-s + (0.0467 − 0.119i)14-s + (−0.475 + 0.823i)16-s + (−0.577 − 1.00i)17-s + 0.0432i·19-s + (1.02 + 0.590i)20-s + (−0.0519 + 0.0899i)22-s + (−0.458 + 0.794i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.111 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.402547 + 0.450195i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.402547 + 0.450195i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (0.393 + 2.61i)T \) |
| 13 | \( 1 + (-1.92 - 3.05i)T \) |
good | 2 | \( 1 + (-0.156 - 0.0904i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (2.32 - 1.34i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 - 2.69iT - 11T^{2} \) |
| 17 | \( 1 + (2.38 + 4.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 0.188iT - 19T^{2} \) |
| 23 | \( 1 + (2.19 - 3.80i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.54 - 6.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.20 - 1.84i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.88 - 3.97i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.70 - 2.71i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.00 - 6.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.60 - 0.924i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.53 - 6.12i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.57 + 3.79i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + 0.411T + 61T^{2} \) |
| 67 | \( 1 - 11.4iT - 67T^{2} \) |
| 71 | \( 1 + (2.89 + 1.67i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (12.3 + 7.10i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.55 + 7.89i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16.5iT - 83T^{2} \) |
| 89 | \( 1 + (-5.10 - 2.94i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.390 - 0.225i)T + (48.5 + 84.0i)T^{2} \) |
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
−10.38596401201177308532843782110, −9.769339987765597886113427398655, −8.864265931145795045803068521780, −7.72501661948623078600139697755, −6.96212539886869578874900742348, −6.33680585962167423336426668254, −4.78231984648752320138900908145, −4.30533656820915359102298317706, −3.21649304547305296061193791882, −1.37718592405877746084852503624,
0.31324479357364569954891513708, 2.56517699515165708017569399260, 3.66193251532385979712271246464, 4.36562940384148497339882393445, 5.49839628487502694352724052664, 6.44415520692179201501315502144, 7.892847266618145146829990754950, 8.435716131118106174933069362171, 8.667248182059317013358143961160, 9.926285298207773674949352348923