L(s) = 1 | + (1.22 + 0.707i)2-s + (0.866 + 1.5i)3-s + (0.999 + 1.73i)4-s + 2.44i·6-s + (6.98 + 0.440i)7-s + 2.82i·8-s + (−1.5 + 2.59i)9-s + (2.76 + 4.79i)11-s + (−1.73 + 2.99i)12-s − 3.50·13-s + (8.24 + 5.47i)14-s + (−2.00 + 3.46i)16-s + (4.06 + 7.04i)17-s + (−3.67 + 2.12i)18-s + (15.3 + 8.86i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 + 0.5i)3-s + (0.249 + 0.433i)4-s + 0.408i·6-s + (0.998 + 0.0628i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.251 + 0.435i)11-s + (−0.144 + 0.249i)12-s − 0.269·13-s + (0.588 + 0.391i)14-s + (−0.125 + 0.216i)16-s + (0.239 + 0.414i)17-s + (−0.204 + 0.117i)18-s + (0.808 + 0.466i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.270 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.395525200\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.395525200\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.22 - 0.707i)T \) |
| 3 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-6.98 - 0.440i)T \) |
good | 11 | \( 1 + (-2.76 - 4.79i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 3.50T + 169T^{2} \) |
| 17 | \( 1 + (-4.06 - 7.04i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-15.3 - 8.86i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (10.3 + 5.96i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 8.52T + 841T^{2} \) |
| 31 | \( 1 + (6.68 - 3.85i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-24.2 - 14.0i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 3.14iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 43.1iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (9.35 - 16.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (2.44 - 1.40i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-26.1 + 15.1i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (41.0 + 23.6i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-5.70 + 3.29i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 97.7T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-30.5 - 52.9i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-61.5 + 106. i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 89.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-102. - 59.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 - 65.1T + 9.40e3T^{2} \) |
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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
−9.954268470676442419492239753892, −9.090484155244774836778748815958, −8.071669774194991317826579301702, −7.65061116946277278514725451377, −6.48461985709450207030463053038, −5.49432773108845586916899948516, −4.71341517225783155984292366447, −3.96858784703203193593195964490, −2.82175999346408411935977408621, −1.58040508197361957577634785982,
0.859644045037976033619192467890, 1.98927342138101932479785944537, 3.04071544560271140772543020563, 4.12291340036270216455184293614, 5.12348411453087967868566799962, 5.89023169129492148492821808361, 7.04225073861119515482081498274, 7.69204128946379521021198014525, 8.625346877868143462802945998474, 9.478657383089860122883483538167