L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.457 + 1.67i)3-s + (0.499 + 0.866i)4-s + (−1.92 − 1.14i)5-s + (0.439 − 1.67i)6-s − 4.60i·7-s − 0.999i·8-s + (−2.58 + 1.52i)9-s + (1.09 + 1.95i)10-s + 6.27i·11-s + (−1.21 + 1.23i)12-s + (1.27 + 2.21i)13-s + (−2.30 + 3.99i)14-s + (1.03 − 3.73i)15-s + (−0.5 + 0.866i)16-s + (−1.20 + 2.08i)17-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.264 + 0.964i)3-s + (0.249 + 0.433i)4-s + (−0.858 − 0.512i)5-s + (0.179 − 0.684i)6-s − 1.74i·7-s − 0.353i·8-s + (−0.860 + 0.509i)9-s + (0.344 + 0.617i)10-s + 1.89i·11-s + (−0.351 + 0.355i)12-s + (0.354 + 0.613i)13-s + (−0.615 + 1.06i)14-s + (0.267 − 0.963i)15-s + (−0.125 + 0.216i)16-s + (−0.292 + 0.506i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00344 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00344 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.534269 + 0.536111i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.534269 + 0.536111i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 + 0.5i)T \) |
| 3 | \( 1 + (-0.457 - 1.67i)T \) |
| 5 | \( 1 + (1.92 + 1.14i)T \) |
| 19 | \( 1 + (-2.80 - 3.33i)T \) |
good | 7 | \( 1 + 4.60iT - 7T^{2} \) |
| 11 | \( 1 - 6.27iT - 11T^{2} \) |
| 13 | \( 1 + (-1.27 - 2.21i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (1.20 - 2.08i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.74 - 3.02i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.557 + 0.965i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.37iT - 31T^{2} \) |
| 37 | \( 1 + 2.27T + 37T^{2} \) |
| 41 | \( 1 + (4.59 - 7.96i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.05 - 2.91i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.24 + 2.14i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0512 - 0.0296i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.22 + 7.31i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.644 + 1.11i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.76 + 3.05i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.26 - 2.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-10.9 - 6.34i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (8.43 + 4.86i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 + (-8.07 - 13.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (3.17 - 5.50i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−10.80874268186776523015825980195, −9.966390647687576404060772457264, −9.502089312173366970479510723702, −8.343180020535874041032217187249, −7.58994010631124662910740824050, −6.86946570693804113187269083397, −4.88178160577845183714527931970, −4.17373376835411946163906532222, −3.49950064398905801912752564069, −1.52052001102488480931805905040,
0.53127644777853632505495187479, 2.54046935087154139503925161393, 3.23798834849036573964457549104, 5.43761434464938648939341036190, 6.08047101502956847322788878724, 7.02722259481792123545820775029, 8.014157875162450466726060332457, 8.687840219387518698485320169312, 9.061165219174079693422408287828, 10.73982150489261010543190487763