L(s) = 1 | + (−0.233 + 0.135i)2-s + (−0.921 + 1.46i)3-s + (−0.963 + 1.66i)4-s + (−1.02 + 0.592i)5-s + (0.0174 − 0.467i)6-s − i·7-s − 1.06i·8-s + (−1.30 − 2.70i)9-s + (0.160 − 0.277i)10-s + (4.19 − 2.42i)11-s + (−1.55 − 2.95i)12-s + (−0.439 − 3.57i)13-s + (0.135 + 0.233i)14-s + (0.0765 − 2.05i)15-s + (−1.78 − 3.08i)16-s + (0.0680 + 0.117i)17-s + ⋯ |
L(s) = 1 | + (−0.165 + 0.0955i)2-s + (−0.531 + 0.846i)3-s + (−0.481 + 0.834i)4-s + (−0.458 + 0.264i)5-s + (0.00713 − 0.190i)6-s − 0.377i·7-s − 0.375i·8-s + (−0.434 − 0.900i)9-s + (0.0506 − 0.0876i)10-s + (1.26 − 0.729i)11-s + (−0.450 − 0.851i)12-s + (−0.121 − 0.992i)13-s + (0.0361 + 0.0625i)14-s + (0.0197 − 0.529i)15-s + (−0.445 − 0.772i)16-s + (0.0165 + 0.0285i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.113i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.769860 - 0.0439601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.769860 - 0.0439601i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.921 - 1.46i)T \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (0.439 + 3.57i)T \) |
good | 2 | \( 1 + (0.233 - 0.135i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (1.02 - 0.592i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.19 + 2.42i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.0680 - 0.117i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.44 + 1.40i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 8.16T + 23T^{2} \) |
| 29 | \( 1 + (-0.654 - 1.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-8.25 + 4.76i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-8.80 - 5.08i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.38iT - 41T^{2} \) |
| 43 | \( 1 + 5.09T + 43T^{2} \) |
| 47 | \( 1 + (5.17 + 2.98i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 0.0422T + 53T^{2} \) |
| 59 | \( 1 + (9.91 + 5.72i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 9.74T + 61T^{2} \) |
| 67 | \( 1 - 4.09iT - 67T^{2} \) |
| 71 | \( 1 + (-6.42 + 3.71i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 8.28iT - 73T^{2} \) |
| 79 | \( 1 + (2.24 - 3.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.79 - 5.07i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3.40 + 1.96i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.5iT - 97T^{2} \) |
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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
−10.00583326392419372512464497269, −9.553950047721550694562432308287, −8.459032159776104166429305841911, −7.86657089602585020936476246767, −6.73073796239777891541796684794, −5.83203326794368353010416085899, −4.60102218904589881651017907364, −3.79934385197003598798659022739, −3.17600517004348269689220917584, −0.52738155159389185754030263135,
1.15933310641280288065267034536, 2.14661593289983955921951072908, 4.17204745129590505331261791172, 4.85540654391382885441844665763, 6.14536412510795458740422853835, 6.49572338035351769767148594422, 7.76689035312912917256049568040, 8.514338438898131706753262307708, 9.531933900098147852066895078602, 10.06978070332413649388067506121