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 | \( 1 + (-0.390 + 0.225i)T + (48.5 - 84.0i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.926285298207773674949352348923, −8.667248182059317013358143961160, −8.435716131118106174933069362171, −7.892847266618145146829990754950, −6.44415520692179201501315502144, −5.49839628487502694352724052664, −4.36562940384148497339882393445, −3.66193251532385979712271246464, −2.56517699515165708017569399260, −0.31324479357364569954891513708,
1.37718592405877746084852503624, 3.21649304547305296061193791882, 4.30533656820915359102298317706, 4.78231984648752320138900908145, 6.33680585962167423336426668254, 6.96212539886869578874900742348, 7.72501661948623078600139697755, 8.864265931145795045803068521780, 9.769339987765597886113427398655, 10.38596401201177308532843782110