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
−10.58473210903519205419259819525, −9.423172467330113019450064214729, −9.088968947717241797720838438072, −7.78474900522153232446424359304, −7.42641379500887011769601821809, −6.41085542090888411122144590157, −5.07930076481082713737341391354, −4.36864964685276662884593438192, −2.87301618836448227733386951655, −0.62485625325438715576426232441,
0.66292751111619430780888216971, 2.45673927399404249464186668486, 3.76048356308522278112803464326, 5.21739530539335118015863492480, 5.84528800073046380757290470504, 7.07323810102618847079437177859, 8.155493020141607089310284027800, 8.719412099996719565843402232565, 9.840305930147426724818338488922, 10.75564291014101645266001186900