L(s) = 1 | + 0.239·2-s − 1.94·4-s + (0.590 + 1.02i)5-s − 0.942·8-s + (0.141 + 0.244i)10-s + (−1.85 + 3.20i)11-s + (−0.5 + 0.866i)13-s + 3.66·16-s + (−3.47 − 6.01i)17-s + (−0.971 + 1.68i)19-s + (−1.14 − 1.98i)20-s + (−0.442 + 0.766i)22-s + (−2.80 − 4.85i)23-s + (1.80 − 3.12i)25-s + (−0.119 + 0.207i)26-s + ⋯ |
L(s) = 1 | + 0.169·2-s − 0.971·4-s + (0.264 + 0.457i)5-s − 0.333·8-s + (0.0446 + 0.0774i)10-s + (−0.558 + 0.967i)11-s + (−0.138 + 0.240i)13-s + 0.915·16-s + (−0.841 − 1.45i)17-s + (−0.222 + 0.385i)19-s + (−0.256 − 0.444i)20-s + (−0.0944 + 0.163i)22-s + (−0.584 − 1.01i)23-s + (0.360 − 0.624i)25-s + (−0.0234 + 0.0406i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.592 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.3996807516\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3996807516\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 5 | \( 1 + (-0.590 - 1.02i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.85 - 3.20i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (3.47 + 6.01i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.971 - 1.68i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.80 + 4.85i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.119 - 0.207i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 1.66T + 31T^{2} \) |
| 37 | \( 1 + (-4.77 + 8.26i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (5.09 - 8.81i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.11 + 1.92i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 5.82T + 47T^{2} \) |
| 53 | \( 1 + (5.80 + 10.0i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 2.60T + 59T^{2} \) |
| 61 | \( 1 + 7.60T + 61T^{2} \) |
| 67 | \( 1 - 3.50T + 67T^{2} \) |
| 71 | \( 1 + 8.60T + 71T^{2} \) |
| 73 | \( 1 + (7.57 + 13.1i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 7.37T + 79T^{2} \) |
| 83 | \( 1 + (3.47 + 6.01i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.37 + 2.37i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.58 + 6.20i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.493726826968732226869271332341, −8.588970060985863520092918046552, −7.76860926900100700568155963327, −6.83757247004473846969537734841, −6.02381670397544633572899976735, −4.76241308872283839482829837903, −4.54026875418306292203550520744, −3.12197714873859518006820076239, −2.12268439234079227724664535969, −0.16271053730768304552844842518,
1.40815270687679638559048349687, 2.99636386068624116479470122842, 3.97201955753257460805180790472, 4.87446421233761338316934978290, 5.65479339761280156888499453767, 6.36123504694371624572985624487, 7.74823409704684600875812991935, 8.446442472321904983377300145714, 8.970273833204323009096298503487, 9.848092085249642689408782925222