L(s) = 1 | + (0.965 − 0.258i)2-s + i·3-s + (0.866 − 0.499i)4-s + (2.82 + 0.756i)5-s + (0.258 + 0.965i)6-s + (−2.36 + 1.17i)7-s + (0.707 − 0.707i)8-s − 9-s + 2.92·10-s + (2.45 − 2.45i)11-s + (0.499 + 0.866i)12-s + (3.36 + 1.28i)13-s + (−1.98 + 1.75i)14-s + (−0.756 + 2.82i)15-s + (0.500 − 0.866i)16-s + (0.623 + 1.07i)17-s + ⋯ |
L(s) = 1 | + (0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (1.26 + 0.338i)5-s + (0.105 + 0.394i)6-s + (−0.895 + 0.445i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s + 0.924·10-s + (0.739 − 0.739i)11-s + (0.144 + 0.249i)12-s + (0.933 + 0.357i)13-s + (−0.529 + 0.468i)14-s + (−0.195 + 0.729i)15-s + (0.125 − 0.216i)16-s + (0.151 + 0.261i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.41377 + 0.647004i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.41377 + 0.647004i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.965 + 0.258i)T \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (2.36 - 1.17i)T \) |
| 13 | \( 1 + (-3.36 - 1.28i)T \) |
good | 5 | \( 1 + (-2.82 - 0.756i)T + (4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (-2.45 + 2.45i)T - 11iT^{2} \) |
| 17 | \( 1 + (-0.623 - 1.07i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.09 - 3.09i)T - 19iT^{2} \) |
| 23 | \( 1 + (1.15 + 0.665i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.896 - 1.55i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.443 - 1.65i)T + (-26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (1.00 + 3.73i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.20 + 0.589i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (4.56 + 2.63i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.98 + 11.1i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (2.96 - 5.14i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.57 - 9.61i)T + (-51.0 - 29.5i)T^{2} \) |
| 61 | \( 1 + 7.76iT - 61T^{2} \) |
| 67 | \( 1 + (2.45 + 2.45i)T + 67iT^{2} \) |
| 71 | \( 1 + (9.20 - 2.46i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (-15.5 + 4.15i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (3.83 + 6.64i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.31 - 2.31i)T - 83iT^{2} \) |
| 89 | \( 1 + (-7.19 + 1.92i)T + (77.0 - 44.5i)T^{2} \) |
| 97 | \( 1 + (2.79 + 10.4i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.69796433164976435601901116008, −10.18163436039498388319284373250, −9.243574721335826213732760627901, −8.553162746585191021825632645361, −6.69768199075128108508867685857, −6.11130760256114212706646953205, −5.52462994645673871356096812576, −3.99737862947852646552187439300, −3.17107227439611556990335150583, −1.86397456283018191194771747945,
1.43253947429979598380912514521, 2.73910250713321341550444650979, 4.05201583609421752317621085395, 5.28085892217441314940721696124, 6.39851604524456816358039108683, 6.57949789859150599229479484993, 7.88331384803777807311307903048, 9.083889814362811649517660407351, 9.787386431245500415667020862424, 10.74732482908756541269106136662