L(s) = 1 | + (−1.56 + 0.900i)2-s + (−0.5 − 0.866i)3-s + (0.623 − 1.07i)4-s − 1.44i·5-s + (1.56 + 0.900i)6-s + (−2.98 − 1.72i)7-s − 1.35i·8-s + (−0.499 + 0.866i)9-s + (1.30 + 2.25i)10-s + (−4.49 + 2.59i)11-s − 1.24·12-s + 6.20·14-s + (−1.25 + 0.722i)15-s + (2.46 + 4.27i)16-s + (−0.376 + 0.652i)17-s − 1.80i·18-s + ⋯ |
L(s) = 1 | + (−1.10 + 0.637i)2-s + (−0.288 − 0.499i)3-s + (0.311 − 0.539i)4-s − 0.646i·5-s + (0.637 + 0.367i)6-s + (−1.12 − 0.651i)7-s − 0.479i·8-s + (−0.166 + 0.288i)9-s + (0.411 + 0.713i)10-s + (−1.35 + 0.781i)11-s − 0.359·12-s + 1.65·14-s + (−0.323 + 0.186i)15-s + (0.617 + 1.06i)16-s + (−0.0913 + 0.158i)17-s − 0.424i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.207 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.198000 + 0.244481i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.198000 + 0.244481i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.56 - 0.900i)T + (1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 1.44iT - 5T^{2} \) |
| 7 | \( 1 + (2.98 + 1.72i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (4.49 - 2.59i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.376 - 0.652i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.89 - 3.98i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.95 - 3.38i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 4.89iT - 31T^{2} \) |
| 37 | \( 1 + (-5.41 + 3.12i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.56 - 0.900i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.54 - 6.14i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 10.5iT - 47T^{2} \) |
| 53 | \( 1 + 3.08T + 53T^{2} \) |
| 59 | \( 1 + (1.62 + 0.939i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.67 - 2.89i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.93 + 2.27i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (7.89 + 4.55i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 2.95iT - 73T^{2} \) |
| 79 | \( 1 + 9.43T + 79T^{2} \) |
| 83 | \( 1 - 6.46iT - 83T^{2} \) |
| 89 | \( 1 + (-1.00 + 0.579i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.49 - 4.32i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.75098135827757529063740367458, −10.01442986304066325518476490204, −9.429693895005883574495883270364, −8.287232703229240312781056406990, −7.59196657999047100187956776272, −6.91382008058073015371233685020, −5.92356702650905268199807043643, −4.70980011831986984914031853630, −3.14992485777983304672731293756, −1.13256160462137661469720063768,
0.32504629097272838986612676365, 2.63226929964739974845627542132, 3.19717104586340876490429707256, 5.15950468799325675328820556119, 5.93728840872067988295654474471, 7.20365600768065254055096817972, 8.262892310366920756824991729634, 9.203120438948985050416373181914, 9.869340557792317176827965824936, 10.43979743715989082986777808361