L(s) = 1 | − 0.239·2-s − 1.94·4-s + (1.29 + 2.24i)5-s + 0.942·8-s + (−0.309 − 0.536i)10-s + (2.09 − 3.62i)11-s + (1.84 − 3.18i)13-s + 3.66·16-s + (−0.855 − 1.48i)17-s + (−3.57 + 6.19i)19-s + (−2.51 − 4.36i)20-s + (−0.499 + 0.866i)22-s + (−2.56 − 4.43i)23-s + (−0.858 + 1.48i)25-s + (−0.440 + 0.762i)26-s + ⋯ |
L(s) = 1 | − 0.169·2-s − 0.971·4-s + (0.579 + 1.00i)5-s + 0.333·8-s + (−0.0979 − 0.169i)10-s + (0.630 − 1.09i)11-s + (0.510 − 0.884i)13-s + 0.915·16-s + (−0.207 − 0.359i)17-s + (−0.820 + 1.42i)19-s + (−0.562 − 0.975i)20-s + (−0.106 + 0.184i)22-s + (−0.534 − 0.925i)23-s + (−0.171 + 0.297i)25-s + (−0.0863 + 0.149i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.990 + 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.359270897\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.359270897\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.29 - 2.24i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.09 + 3.62i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.84 + 3.18i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.855 + 1.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.57 - 6.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.56 + 4.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.06 + 1.84i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 6.53T + 31T^{2} \) |
| 37 | \( 1 + (0.830 - 1.43i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.10 + 8.84i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.830 - 1.43i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 9.33T + 47T^{2} \) |
| 53 | \( 1 + (-5.32 - 9.22i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 6.06T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 - 8.26T + 67T^{2} \) |
| 71 | \( 1 + 6.23T + 71T^{2} \) |
| 73 | \( 1 + (-3.57 - 6.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 9.82T + 79T^{2} \) |
| 83 | \( 1 + (3.44 + 5.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-2.51 + 4.36i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.53 - 2.65i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.732542722811578622021756382215, −8.606435407997628234348004786083, −8.339431259208018095630720291033, −7.16978722489079055738386121508, −5.95869130905029225642879229766, −5.85915777441183649221252643714, −4.30562496055761884684906446727, −3.55884517963912215635037956991, −2.45040554146699242083619653907, −0.808701550195423278292873538607,
1.04906529465383329021174136556, 2.10777827890589239428074877606, 3.94166799161481498037360584522, 4.51451739407715670514577839038, 5.26346491073892705531290051054, 6.31439311776678634619580011132, 7.21586598779652157668316897120, 8.395327504254487083179602055574, 8.930940593749906622373306619903, 9.484442781456852256614296232815