L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (0.499 + 0.866i)6-s + (2.59 − 1.5i)7-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−0.5 + 0.866i)11-s + 0.999i·12-s + (2.59 + 2.5i)13-s + 3·14-s + (−0.5 + 0.866i)16-s + 0.999i·18-s + (−2.5 − 4.33i)19-s + 3·21-s + (−0.866 + 0.499i)22-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 + 0.288i)3-s + (0.249 + 0.433i)4-s + (0.204 + 0.353i)6-s + (0.981 − 0.566i)7-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.150 + 0.261i)11-s + 0.288i·12-s + (0.720 + 0.693i)13-s + 0.801·14-s + (−0.125 + 0.216i)16-s + 0.235i·18-s + (−0.573 − 0.993i)19-s + 0.654·21-s + (−0.184 + 0.106i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.540 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.490647573\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.490647573\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 + (-2.59 - 2.5i)T \) |
good | 7 | \( 1 + (-2.59 + 1.5i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.5 + 4.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 10T + 31T^{2} \) |
| 37 | \( 1 + (0.866 + 0.5i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3 - 5.19i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (1.73 - i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 9iT - 47T^{2} \) |
| 53 | \( 1 + 13iT - 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (10.3 + 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 16iT - 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.3 - 6i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.135102954329212783389807679659, −8.391468159437167466493013009523, −7.78591773425927211742643925153, −6.91744708108375271935424553366, −6.22787042431572163422041336642, −4.85568446522594580590303025835, −4.63211044674775159380957326690, −3.63479778008292596754054398793, −2.59444830673061003757048273862, −1.41013736048468723594731439097,
1.13994476464709636524394066693, 2.18209457349001252824412030441, 3.07827941879822769644181305241, 4.03341269987190224942214397921, 4.98649936128342067461450213170, 5.76738666415937700560585516389, 6.54173596906625829547403904917, 7.60931145085447935601592650276, 8.453760671093090757601273565914, 8.722977334353497959341185837325