L(s) = 1 | + 0.180i·2-s + (0.913 − 1.58i)3-s + 1.96·4-s + (−2.32 − 1.34i)5-s + (0.285 + 0.165i)6-s + (−2.46 + 0.967i)7-s + 0.717i·8-s + (−0.167 − 0.289i)9-s + (0.242 − 0.420i)10-s + (2.33 + 1.34i)11-s + (1.79 − 3.11i)12-s + (1.92 + 3.05i)13-s + (−0.174 − 0.445i)14-s + (−4.24 + 2.45i)15-s + 3.80·16-s − 4.76·17-s + ⋯ |
L(s) = 1 | + 0.127i·2-s + (0.527 − 0.913i)3-s + 0.983·4-s + (−1.04 − 0.600i)5-s + (0.116 + 0.0673i)6-s + (−0.930 + 0.365i)7-s + 0.253i·8-s + (−0.0557 − 0.0965i)9-s + (0.0768 − 0.133i)10-s + (0.703 + 0.406i)11-s + (0.518 − 0.898i)12-s + (0.532 + 0.846i)13-s + (−0.0467 − 0.119i)14-s + (−1.09 + 0.633i)15-s + 0.951·16-s − 1.15·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.834 + 0.551i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08657 - 0.326583i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08657 - 0.326583i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.46 - 0.967i)T \) |
| 13 | \( 1 + (-1.92 - 3.05i)T \) |
good | 2 | \( 1 - 0.180iT - 2T^{2} \) |
| 3 | \( 1 + (-0.913 + 1.58i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.32 + 1.34i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.33 - 1.34i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + 4.76T + 17T^{2} \) |
| 19 | \( 1 + (-0.163 + 0.0942i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 4.39T + 23T^{2} \) |
| 29 | \( 1 + (3.54 + 6.13i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.20 - 1.84i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 7.95iT - 37T^{2} \) |
| 41 | \( 1 + (-4.70 + 2.71i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.00 - 6.93i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.60 + 0.924i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.53 - 6.12i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 7.58iT - 59T^{2} \) |
| 61 | \( 1 + (-0.205 - 0.356i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.87 + 5.70i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.89 - 1.67i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-12.3 + 7.10i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.55 - 7.89i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16.5iT - 83T^{2} \) |
| 89 | \( 1 - 5.89iT - 89T^{2} \) |
| 97 | \( 1 + (-0.390 - 0.225i)T + (48.5 + 84.0i)T^{2} \) |
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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
−13.90518716341126251498465173057, −12.73310595999314207728334165161, −12.07842933616434574852906421927, −11.11804390652511524314278579675, −9.320962546863637723493971751891, −8.186117707355295235776713678945, −7.16024655789749945043547252587, −6.28199847173491619669057323411, −3.97848288246132326619559631352, −2.10531560573797820735164018391,
3.20470804234942167004027587377, 3.86502760398039012710136254043, 6.29794709352048733301797456550, 7.31544515331466757826428984552, 8.726738667998116797728923067912, 10.06895726017323947839011095502, 10.88499545421245622198503154088, 11.80567285566522508282262580131, 13.09505747844425887261813534255, 14.58768272957122665469205271387