L(s) = 1 | + (0.127 + 0.474i)2-s + (1.58 − 0.708i)3-s + (1.52 − 0.879i)4-s + (−1.06 − 1.96i)5-s + (0.536 + 0.659i)6-s + (1.30 + 1.30i)8-s + (1.99 − 2.24i)9-s + (0.795 − 0.755i)10-s + (2.31 − 1.33i)11-s + (1.78 − 2.46i)12-s + (−2.14 + 2.14i)13-s + (−3.07 − 2.34i)15-s + (1.30 − 2.26i)16-s + (−4.46 − 1.19i)17-s + (1.31 + 0.661i)18-s + (4.54 + 2.62i)19-s + ⋯ |
L(s) = 1 | + (0.0898 + 0.335i)2-s + (0.912 − 0.409i)3-s + (0.761 − 0.439i)4-s + (−0.477 − 0.878i)5-s + (0.219 + 0.269i)6-s + (0.461 + 0.461i)8-s + (0.665 − 0.746i)9-s + (0.251 − 0.238i)10-s + (0.697 − 0.402i)11-s + (0.515 − 0.712i)12-s + (−0.596 + 0.596i)13-s + (−0.795 − 0.606i)15-s + (0.326 − 0.565i)16-s + (−1.08 − 0.290i)17-s + (0.310 + 0.155i)18-s + (1.04 + 0.601i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.628 + 0.778i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.628 + 0.778i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.21811 - 1.06012i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.21811 - 1.06012i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.58 + 0.708i)T \) |
| 5 | \( 1 + (1.06 + 1.96i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.127 - 0.474i)T + (-1.73 + i)T^{2} \) |
| 11 | \( 1 + (-2.31 + 1.33i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.14 - 2.14i)T - 13iT^{2} \) |
| 17 | \( 1 + (4.46 + 1.19i)T + (14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (-4.54 - 2.62i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.48 - 0.932i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 2.86T + 29T^{2} \) |
| 31 | \( 1 + (2.64 + 4.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.92 + 0.784i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 - 11.5iT - 41T^{2} \) |
| 43 | \( 1 + (-0.759 + 0.759i)T - 43iT^{2} \) |
| 47 | \( 1 + (2.80 + 10.4i)T + (-40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (1.62 - 6.05i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (0.0797 + 0.138i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.36 + 4.09i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (1.98 - 7.39i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 - 13.5iT - 71T^{2} \) |
| 73 | \( 1 + (-5.68 - 1.52i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (3.37 + 1.94i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.03 - 4.03i)T + 83iT^{2} \) |
| 89 | \( 1 + (1.97 - 3.42i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.86 - 1.86i)T + 97iT^{2} \) |
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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
−9.953553059744547397775748405855, −9.336041604013607484731276929788, −8.407128214495595795550594704780, −7.65464509549854232511467637066, −6.89752142218393925000407295749, −5.98977113317056953122564410631, −4.76146271593818886539002940357, −3.74687118788334083503696456647, −2.35021529137552103239788438120, −1.22689464464737416397871378944,
2.02632625316240747212084670365, 2.93236893220651445807558207802, 3.71729281310738522385676697097, 4.67828927106037726899802445870, 6.39424577858755610819596007155, 7.23894018924505071261624085332, 7.75014910261249361194108812812, 8.792454498842436720453762189541, 9.803206978510878870997088104807, 10.55230189509729335305647757917