L(s) = 1 | + (0.342 − 0.939i)2-s + (2.25 − 0.397i)3-s + (−0.766 − 0.642i)4-s + (0.397 − 2.25i)6-s + (−2.39 − 1.38i)7-s + (−0.866 + 0.500i)8-s + (2.10 − 0.764i)9-s + (−1.21 − 2.09i)11-s + (−1.98 − 1.14i)12-s + (1.70 + 0.300i)13-s + (−2.12 + 1.77i)14-s + (0.173 + 0.984i)16-s + (1.38 − 3.79i)17-s − 2.23i·18-s + (−0.0664 − 4.35i)19-s + ⋯ |
L(s) = 1 | + (0.241 − 0.664i)2-s + (1.30 − 0.229i)3-s + (−0.383 − 0.321i)4-s + (0.162 − 0.920i)6-s + (−0.906 − 0.523i)7-s + (−0.306 + 0.176i)8-s + (0.700 − 0.254i)9-s + (−0.365 − 0.632i)11-s + (−0.572 − 0.330i)12-s + (0.472 + 0.0833i)13-s + (−0.566 + 0.475i)14-s + (0.0434 + 0.246i)16-s + (0.334 − 0.919i)17-s − 0.527i·18-s + (−0.0152 − 0.999i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.614 + 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.967424 - 1.98019i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.967424 - 1.98019i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.342 + 0.939i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (0.0664 + 4.35i)T \) |
good | 3 | \( 1 + (-2.25 + 0.397i)T + (2.81 - 1.02i)T^{2} \) |
| 7 | \( 1 + (2.39 + 1.38i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.21 + 2.09i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.70 - 0.300i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.38 + 3.79i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-5.74 + 6.84i)T + (-3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (4.18 - 1.52i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-0.953 + 1.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 2.89iT - 37T^{2} \) |
| 41 | \( 1 + (0.808 + 4.58i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-1.91 - 2.28i)T + (-7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-3.39 - 9.33i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (7.18 - 8.56i)T + (-9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.89 - 0.689i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-6.66 - 5.59i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-4.61 - 12.6i)T + (-51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-10.2 + 8.57i)T + (12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-5.62 + 0.991i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (1.16 + 6.62i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-14.6 - 8.47i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.964 - 5.46i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-2.63 + 7.22i)T + (-74.3 - 62.3i)T^{2} \) |
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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
−9.507392279542907889464457115614, −9.110011846178804106857359502745, −8.265432479500359324841608008199, −7.29538482618196360050341665841, −6.44541516922020786543263519424, −5.16224773161116839363474128737, −3.98648541135256075987222859320, −3.05324620881851238797022556990, −2.57133285675857838917808516294, −0.819811112208797898434441412247,
2.03073137192104623407562468810, 3.35935392667852767861410418420, 3.72944707134270429461148335947, 5.20746380668361906843142543271, 6.04828009618461869306934889252, 7.06833044381450770437917050033, 7.967496489368814919745922044735, 8.524528592777577589678294187936, 9.521912135136195883091839300802, 9.770640397108549256620677245326