L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (−0.499 + 0.866i)6-s + (1.73 + i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1 − 1.73i)11-s − 0.999i·12-s + (−2.59 − 2.5i)13-s − 1.99·14-s + (−0.5 − 0.866i)16-s + (−4.33 − 2.5i)17-s + 0.999i·18-s + (−1 + 1.73i)19-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (−0.204 + 0.353i)6-s + (0.654 + 0.377i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.301 − 0.522i)11-s − 0.288i·12-s + (−0.720 − 0.693i)13-s − 0.534·14-s + (−0.125 − 0.216i)16-s + (−1.05 − 0.606i)17-s + 0.235i·18-s + (−0.229 + 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.572i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4615745325\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4615745325\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (2.59 + 2.5i)T \) |
good | 7 | \( 1 + (-1.73 - i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1 + 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.33 + 2.5i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (9.52 - 5.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.66 - 5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2iT - 47T^{2} \) |
| 53 | \( 1 - iT - 53T^{2} \) |
| 59 | \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 + i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 13iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.73 - i)T + (48.5 + 84.0i)T^{2} \) |
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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
−8.781419066586178115389846856731, −7.968186421018230414667153179630, −7.65706384904069473983801997296, −6.64702588859483034623984913483, −5.73946222763695844080875234188, −5.01582029689882105670062074960, −3.81435204865125962142531271564, −2.56215244996036002473197197903, −1.80728138493685538605414544404, −0.17225415088682102828990645716,
1.78066729123762837404490369090, 2.33228084191084911806414021304, 3.73037395961379360625968847852, 4.41235510107033913759963203982, 5.30218831435858337224796567558, 6.71252635208430678622137515276, 7.29096736512624528617977635040, 8.071214113707389444054169378377, 8.878152185997758900313546834796, 9.327299271955976713581550494189