L(s) = 1 | + (1.91 − 2.63i)2-s + (−2.03 − 6.26i)4-s + (0.936 + 1.28i)5-s + (−3.36 − 10.3i)7-s + (−8.01 − 2.60i)8-s + 5.18·10-s + (5.25 + 9.66i)11-s + (7.47 + 5.43i)13-s + (−33.6 − 10.9i)14-s + (−0.868 + 0.630i)16-s + (2.83 + 3.89i)17-s + (−7.75 + 23.8i)19-s + (6.17 − 8.49i)20-s + (35.4 + 4.64i)22-s + 36.6i·23-s + ⋯ |
L(s) = 1 | + (0.956 − 1.31i)2-s + (−0.509 − 1.56i)4-s + (0.187 + 0.257i)5-s + (−0.480 − 1.47i)7-s + (−1.00 − 0.325i)8-s + 0.518·10-s + (0.477 + 0.878i)11-s + (0.574 + 0.417i)13-s + (−2.40 − 0.781i)14-s + (−0.0542 + 0.0394i)16-s + (0.166 + 0.229i)17-s + (−0.408 + 1.25i)19-s + (0.308 − 0.424i)20-s + (1.61 + 0.211i)22-s + 1.59i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.14001 - 1.73414i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14001 - 1.73414i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 + (-5.25 - 9.66i)T \) |
good | 2 | \( 1 + (-1.91 + 2.63i)T + (-1.23 - 3.80i)T^{2} \) |
| 5 | \( 1 + (-0.936 - 1.28i)T + (-7.72 + 23.7i)T^{2} \) |
| 7 | \( 1 + (3.36 + 10.3i)T + (-39.6 + 28.8i)T^{2} \) |
| 13 | \( 1 + (-7.47 - 5.43i)T + (52.2 + 160. i)T^{2} \) |
| 17 | \( 1 + (-2.83 - 3.89i)T + (-89.3 + 274. i)T^{2} \) |
| 19 | \( 1 + (7.75 - 23.8i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 - 36.6iT - 529T^{2} \) |
| 29 | \( 1 + (-23.2 + 7.55i)T + (680. - 494. i)T^{2} \) |
| 31 | \( 1 + (47.0 + 34.1i)T + (296. + 913. i)T^{2} \) |
| 37 | \( 1 + (-3.95 - 12.1i)T + (-1.10e3 + 804. i)T^{2} \) |
| 41 | \( 1 + (-11.8 - 3.83i)T + (1.35e3 + 988. i)T^{2} \) |
| 43 | \( 1 - 19.6T + 1.84e3T^{2} \) |
| 47 | \( 1 + (61.7 + 20.0i)T + (1.78e3 + 1.29e3i)T^{2} \) |
| 53 | \( 1 + (29.0 - 40.0i)T + (-868. - 2.67e3i)T^{2} \) |
| 59 | \( 1 + (22.9 - 7.44i)T + (2.81e3 - 2.04e3i)T^{2} \) |
| 61 | \( 1 + (35.7 - 26.0i)T + (1.14e3 - 3.53e3i)T^{2} \) |
| 67 | \( 1 - 33.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + (44.0 + 60.6i)T + (-1.55e3 + 4.79e3i)T^{2} \) |
| 73 | \( 1 + (-12.7 - 39.3i)T + (-4.31e3 + 3.13e3i)T^{2} \) |
| 79 | \( 1 + (51.7 + 37.6i)T + (1.92e3 + 5.93e3i)T^{2} \) |
| 83 | \( 1 + (-24.7 - 34.1i)T + (-2.12e3 + 6.55e3i)T^{2} \) |
| 89 | \( 1 + 111. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-87.4 - 63.5i)T + (2.90e3 + 8.94e3i)T^{2} \) |
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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
−13.26530223746762277375448924307, −12.34855244305636733787507532045, −11.26275779997334529195064003425, −10.32750325395227327457213289257, −9.645458607497048829654683027484, −7.53503927833092994326898819635, −6.13524541696952821807882658123, −4.34617077703888230162413013035, −3.55842824503154296302468457943, −1.59643275233536029837711212027,
3.16407363829547859725488901342, 4.94795846183929983372630026424, 5.92035219551504672552637412988, 6.75917058242658046783512692631, 8.445644501822269702863817600541, 9.065645949730488301168136279478, 11.02570489352898833612459180399, 12.48114284761944748557698745978, 13.01071914421567444901266890411, 14.19690392852803004310089866350