L(s) = 1 | + (0.384 + 0.666i)2-s + (−2.14 − 0.779i)3-s + (0.703 − 1.21i)4-s + (1.33 − 1.11i)5-s + (−0.304 − 1.72i)6-s + (0.847 − 0.711i)7-s + 2.62·8-s + (1.68 + 1.41i)9-s + (1.25 + 0.457i)10-s + (−0.455 + 0.165i)11-s + (−2.45 + 2.06i)12-s + (−1.20 + 1.01i)13-s + (0.800 + 0.291i)14-s + (−3.72 + 1.35i)15-s + (−0.398 − 0.690i)16-s + (1.52 + 2.63i)17-s + ⋯ |
L(s) = 1 | + (0.272 + 0.471i)2-s + (−1.23 − 0.450i)3-s + (0.351 − 0.609i)4-s + (0.595 − 0.499i)5-s + (−0.124 − 0.705i)6-s + (0.320 − 0.268i)7-s + 0.927·8-s + (0.560 + 0.470i)9-s + (0.397 + 0.144i)10-s + (−0.137 + 0.0500i)11-s + (−0.709 + 0.595i)12-s + (−0.334 + 0.280i)13-s + (0.213 + 0.0778i)14-s + (−0.961 + 0.349i)15-s + (−0.0996 − 0.172i)16-s + (0.368 + 0.639i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 109 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.952060 - 0.282672i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.952060 - 0.282672i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 109 | \( 1 + (4.96 + 9.18i)T \) |
good | 2 | \( 1 + (-0.384 - 0.666i)T + (-1 + 1.73i)T^{2} \) |
| 3 | \( 1 + (2.14 + 0.779i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-1.33 + 1.11i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.847 + 0.711i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (0.455 - 0.165i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (1.20 - 1.01i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.52 - 2.63i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 + 2.44i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.40 - 5.88i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-9.24 - 3.36i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (5.98 + 5.02i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.99 - 2.51i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + 0.837T + 41T^{2} \) |
| 43 | \( 1 + (-0.543 - 0.941i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.25 - 7.14i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-3.70 + 3.10i)T + (9.20 - 52.1i)T^{2} \) |
| 59 | \( 1 + (-1.51 + 0.552i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (2.44 + 13.8i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (11.0 + 9.25i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.54 + 2.68i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.45 - 2.35i)T + (55.9 - 46.9i)T^{2} \) |
| 79 | \( 1 + (-10.8 - 9.09i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.50 + 0.911i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (2.36 + 13.3i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (1.51 - 1.26i)T + (16.8 - 95.5i)T^{2} \) |
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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
−13.58131506935203673400265372032, −12.58566046191863217902228732461, −11.47345213889979344343693410613, −10.65035641645083176573102091054, −9.512095852896473796445619418212, −7.69168087572007604559803206004, −6.51928663894432697277073329861, −5.68172656336782479474723788061, −4.77669608542845920936960296704, −1.49550901784354770057515034259,
2.55974529156420730716563989668, 4.41559310642913391204668294016, 5.67779170850501144519689493780, 6.83574448345396867683976164530, 8.305532509889948676090251631015, 10.22229836598353414151971316230, 10.60148748389778623325196532480, 11.85730682629471850854978367297, 12.25918638166774476422101782256, 13.64362807891741973523475597683