L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + 0.999·6-s + (−0.866 − 2.5i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−3 − 5.19i)11-s + (0.866 + 0.499i)12-s − 4i·13-s + (0.500 − 2.59i)14-s + (−0.5 + 0.866i)16-s + (2.59 − 1.5i)17-s + (0.866 − 0.499i)18-s + (−2 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + 0.408·6-s + (−0.327 − 0.944i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.904 − 1.56i)11-s + (0.249 + 0.144i)12-s − 1.10i·13-s + (0.133 − 0.694i)14-s + (−0.125 + 0.216i)16-s + (0.630 − 0.363i)17-s + (0.204 − 0.117i)18-s + (−0.458 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.441 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.282919946\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.282919946\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (0.866 + 2.5i)T \) |
good | 11 | \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 + (-2.59 + 1.5i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (2.5 + 4.33i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 3T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + (-7.79 - 4.5i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.3 + 6i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3 - 5.19i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.92 - 4i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 9T + 71T^{2} \) |
| 73 | \( 1 + (-12.1 + 7i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.5 - 6.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-1.5 + 2.59i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 17iT - 97T^{2} \) |
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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
−9.964036705942057028807875038888, −8.552923917569834415450490119327, −7.988172662892135423169909264164, −7.40921036504126359539888414805, −6.19804490651383943895751881021, −5.68359393025157227187238677789, −4.40322265965327691128932082319, −3.37970273041077530920096457158, −2.74456253382226711071637230133, −0.77563940836790596139122098489,
2.00394974214287043147633418900, 2.60013347672798769382334623438, 3.88629644280544625628283227217, 4.74490855693351358488029120181, 5.54145640256597988457700014120, 6.67495449620542958797644509197, 7.46088642320246444504857177966, 8.602239988955811157661525439501, 9.345499590436660810476807158840, 10.06783087852723639595273511492