L(s) = 1 | + (−1.10 − 0.638i)2-s − 1.16·3-s + (−0.185 − 0.320i)4-s + (−1.57 + 0.907i)5-s + (1.29 + 0.745i)6-s + 3.02i·8-s − 1.63·9-s + 2.31·10-s − 2.77i·11-s + (0.216 + 0.374i)12-s + (−3.58 − 0.402i)13-s + (1.83 − 1.05i)15-s + (1.56 − 2.70i)16-s + (1.37 + 2.37i)17-s + (1.80 + 1.04i)18-s + 5.86i·19-s + ⋯ |
L(s) = 1 | + (−0.781 − 0.451i)2-s − 0.674·3-s + (−0.0925 − 0.160i)4-s + (−0.702 + 0.405i)5-s + (0.527 + 0.304i)6-s + 1.06i·8-s − 0.545·9-s + 0.732·10-s − 0.837i·11-s + (0.0624 + 0.108i)12-s + (−0.993 − 0.111i)13-s + (0.473 − 0.273i)15-s + (0.390 − 0.675i)16-s + (0.332 + 0.576i)17-s + (0.426 + 0.246i)18-s + 1.34i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.867 + 0.498i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.422464 - 0.112710i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.422464 - 0.112710i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 13 | \( 1 + (3.58 + 0.402i)T \) |
good | 2 | \( 1 + (1.10 + 0.638i)T + (1 + 1.73i)T^{2} \) |
| 3 | \( 1 + 1.16T + 3T^{2} \) |
| 5 | \( 1 + (1.57 - 0.907i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + 2.77iT - 11T^{2} \) |
| 17 | \( 1 + (-1.37 - 2.37i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 - 5.86iT - 19T^{2} \) |
| 23 | \( 1 + (-3.49 + 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.75 - 3.04i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (1.79 + 1.03i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.50 + 0.871i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.51 + 3.18i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.55 + 7.88i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.76 + 3.32i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.24 + 9.08i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.66 - 1.53i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 - 5.01iT - 67T^{2} \) |
| 71 | \( 1 + (-2.35 - 1.35i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.64 - 3.83i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.86 - 13.6i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 + (-13.9 - 8.03i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (12.3 + 7.11i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.75564291014101645266001186900, −9.840305930147426724818338488922, −8.719412099996719565843402232565, −8.155493020141607089310284027800, −7.07323810102618847079437177859, −5.84528800073046380757290470504, −5.21739530539335118015863492480, −3.76048356308522278112803464326, −2.45673927399404249464186668486, −0.66292751111619430780888216971,
0.62485625325438715576426232441, 2.87301618836448227733386951655, 4.36864964685276662884593438192, 5.07930076481082713737341391354, 6.41085542090888411122144590157, 7.42641379500887011769601821809, 7.78474900522153232446424359304, 9.088968947717241797720838438072, 9.423172467330113019450064214729, 10.58473210903519205419259819525