L(s) = 1 | + (−0.292 + 0.292i)2-s + (−1 − 1.41i)3-s + 1.82i·4-s + (−2 + i)5-s + (0.707 + 0.121i)6-s + (−3.41 − 3.41i)7-s + (−1.12 − 1.12i)8-s + (−1.00 + 2.82i)9-s + (0.292 − 0.878i)10-s − i·11-s + (2.58 − 1.82i)12-s + (−2 + 2i)13-s + 2·14-s + (3.41 + 1.82i)15-s − 3·16-s + (−2.82 + 2.82i)17-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.207i)2-s + (−0.577 − 0.816i)3-s + 0.914i·4-s + (−0.894 + 0.447i)5-s + (0.288 + 0.0495i)6-s + (−1.29 − 1.29i)7-s + (−0.396 − 0.396i)8-s + (−0.333 + 0.942i)9-s + (0.0926 − 0.277i)10-s − 0.301i·11-s + (0.746 − 0.527i)12-s + (−0.554 + 0.554i)13-s + 0.534·14-s + (0.881 + 0.472i)15-s − 0.750·16-s + (−0.685 + 0.685i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 165 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1 + 1.41i)T \) |
| 5 | \( 1 + (2 - i)T \) |
| 11 | \( 1 + iT \) |
good | 2 | \( 1 + (0.292 - 0.292i)T - 2iT^{2} \) |
| 7 | \( 1 + (3.41 + 3.41i)T + 7iT^{2} \) |
| 13 | \( 1 + (2 - 2i)T - 13iT^{2} \) |
| 17 | \( 1 + (2.82 - 2.82i)T - 17iT^{2} \) |
| 19 | \( 1 + 2.82iT - 19T^{2} \) |
| 23 | \( 1 + (-3.24 - 3.24i)T + 23iT^{2} \) |
| 29 | \( 1 - 0.828T + 29T^{2} \) |
| 31 | \( 1 + 3.17T + 31T^{2} \) |
| 37 | \( 1 + (0.171 + 0.171i)T + 37iT^{2} \) |
| 41 | \( 1 + 7.65iT - 41T^{2} \) |
| 43 | \( 1 + (-0.242 + 0.242i)T - 43iT^{2} \) |
| 47 | \( 1 + (7.24 - 7.24i)T - 47iT^{2} \) |
| 53 | \( 1 + (-1 - i)T + 53iT^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 6.48T + 61T^{2} \) |
| 67 | \( 1 + (6.41 + 6.41i)T + 67iT^{2} \) |
| 71 | \( 1 + 2.48iT - 71T^{2} \) |
| 73 | \( 1 - 73iT^{2} \) |
| 79 | \( 1 - 10.8iT - 79T^{2} \) |
| 83 | \( 1 + (9.07 + 9.07i)T + 83iT^{2} \) |
| 89 | \( 1 + 9.65T + 89T^{2} \) |
| 97 | \( 1 + (-10.6 - 10.6i)T + 97iT^{2} \) |
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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
−12.46444367443615871812057143837, −11.40586678527199925533501998430, −10.62454172024670993093780973925, −9.114942403298047233521329827191, −7.76294528889955773847644198854, −7.07300974228222624945742134198, −6.50134671005910394776777764714, −4.28489111140961153529046010006, −3.09767553925875274984580276236, 0,
2.98891102029091138630765488653, 4.71378656126310280001614070861, 5.64188308833732431920583506532, 6.73110362974399645876515485018, 8.681644775252795112526704394372, 9.420057106147017479690307448405, 10.19135293293924046129024827951, 11.34900119018502978671986623422, 12.12159864060428824165989212479