L(s) = 1 | + (1.80 + 2.48i)2-s + (−1.69 + 5.20i)4-s + (3.23 + 2.35i)5-s + (−0.854 − 0.277i)7-s + (−4.30 + 1.40i)8-s + 12.3i·10-s + (−10.8 + 1.67i)11-s + (−5 − 6.88i)13-s + (−0.854 − 2.62i)14-s + (6.42 + 4.66i)16-s + (14.5 − 20.0i)17-s + (11.2 − 3.66i)19-s + (−17.7 + 12.8i)20-s + (−23.8 − 24.0i)22-s + 7.23·23-s + ⋯ |
L(s) = 1 | + (0.904 + 1.24i)2-s + (−0.422 + 1.30i)4-s + (0.647 + 0.470i)5-s + (−0.122 − 0.0396i)7-s + (−0.538 + 0.175i)8-s + 1.23i·10-s + (−0.988 + 0.152i)11-s + (−0.384 − 0.529i)13-s + (−0.0610 − 0.187i)14-s + (0.401 + 0.291i)16-s + (0.854 − 1.17i)17-s + (0.593 − 0.192i)19-s + (−0.885 + 0.643i)20-s + (−1.08 − 1.09i)22-s + 0.314·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.33257 + 1.66798i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33257 + 1.66798i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (10.8 - 1.67i)T \) |
good | 2 | \( 1 + (-1.80 - 2.48i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (-3.23 - 2.35i)T + (7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (0.854 + 0.277i)T + (39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (5 + 6.88i)T + (-52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-14.5 + 20.0i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (-11.2 + 3.66i)T + (292. - 212. i)T^{2} \) |
| 23 | \( 1 - 7.23T + 529T^{2} \) |
| 29 | \( 1 + (-3.29 - 1.06i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (26.7 - 19.4i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-12.4 + 38.2i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (1.24 - 0.403i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 - 33.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (7.03 + 21.6i)T + (-1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (63.5 - 46.1i)T + (868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (9.66 - 29.7i)T + (-2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (16.5 - 22.7i)T + (-1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 + 76.5T + 4.48e3T^{2} \) |
| 71 | \( 1 + (50.4 + 36.6i)T + (1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-89.5 - 29.0i)T + (4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (-38.7 - 53.3i)T + (-1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (33.3 - 45.8i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 - 62.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + (58.5 - 42.5i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−14.08739050996393998526976790028, −13.28961050512169790650234246457, −12.32681099229582438818423066625, −10.67874275075683546028755532021, −9.600623639454218331680525822802, −7.86208297219498821027088470108, −7.06383149669678687868769968854, −5.76992838221679031801503041478, −4.94340296374569361003932209293, −3.01650031161770013409955576841,
1.73612757667174877738344866303, 3.29360658511120770333031602520, 4.84779648901546311985560239426, 5.84168107808813450883461116481, 7.84194993958513371589912902940, 9.478187163293610985764691510454, 10.32869945961975639359311238390, 11.38569573412292121873336776484, 12.51195060785668633618949744546, 13.10021035991964999543296557659