L(s) = 1 | + (0.288 + 0.500i)2-s + (−1.33 + 1.10i)3-s + (0.833 − 1.44i)4-s + (−1.50 + 2.60i)5-s + (−0.938 − 0.346i)6-s + (−0.582 − 1.00i)7-s + 2.11·8-s + (0.549 − 2.94i)9-s − 1.73·10-s + (0.487 + 2.84i)12-s + (−1.97 + 3.41i)13-s + (0.336 − 0.582i)14-s + (−0.878 − 5.12i)15-s + (−1.05 − 1.82i)16-s + 0.314·17-s + (1.63 − 0.577i)18-s + ⋯ |
L(s) = 1 | + (0.204 + 0.353i)2-s + (−0.769 + 0.639i)3-s + (0.416 − 0.721i)4-s + (−0.671 + 1.16i)5-s + (−0.383 − 0.141i)6-s + (−0.220 − 0.381i)7-s + 0.748·8-s + (0.183 − 0.983i)9-s − 0.548·10-s + (0.140 + 0.821i)12-s + (−0.547 + 0.947i)13-s + (0.0899 − 0.155i)14-s + (−0.226 − 1.32i)15-s + (−0.263 − 0.456i)16-s + 0.0762·17-s + (0.385 − 0.136i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 + 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.05129403207\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05129403207\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.33 - 1.10i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.288 - 0.500i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.50 - 2.60i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.582 + 1.00i)T + (-3.5 + 6.06i)T^{2} \) |
| 13 | \( 1 + (1.97 - 3.41i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.314T + 17T^{2} \) |
| 19 | \( 1 + 6.36T + 19T^{2} \) |
| 23 | \( 1 + (-0.0427 + 0.0740i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.84 + 6.66i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.26 - 5.65i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 6.24T + 37T^{2} \) |
| 41 | \( 1 + (-2.90 + 5.03i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.39 + 5.87i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.163 - 0.283i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 2.42T + 53T^{2} \) |
| 59 | \( 1 + (-1.14 + 1.98i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (6.84 + 11.8i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.83 - 10.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.71T + 71T^{2} \) |
| 73 | \( 1 + 2.94T + 73T^{2} \) |
| 79 | \( 1 + (4.44 + 7.69i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.47 - 4.29i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 2.12T + 89T^{2} \) |
| 97 | \( 1 + (-0.0444 - 0.0769i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.940031801558497907717341490978, −8.927136786846628520713550711751, −7.49386048282740185458726269228, −6.82672402902451767966504144357, −6.39757754607473742112720351561, −5.35325037842733590326060487323, −4.35881313806542406054306071503, −3.59739259790607516227275768346, −2.08393271357974686796692496454, −0.02230637780709788319074858518,
1.57560366500299906893430854887, 2.79613512571007658130531343446, 4.10545943725699604368381904952, 4.90679816472410727695639110983, 5.82771190584330195973508530715, 6.88504512326668195821662908204, 7.76156131635318872968808278433, 8.243148002941783865781418236560, 9.172661149559523146178596976561, 10.52117298936551165820796759693