| 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
−13.64362807891741973523475597683, −12.25918638166774476422101782256, −11.85730682629471850854978367297, −10.60148748389778623325196532480, −10.22229836598353414151971316230, −8.305532509889948676090251631015, −6.83574448345396867683976164530, −5.67779170850501144519689493780, −4.41559310642913391204668294016, −2.55974529156420730716563989668,
1.49550901784354770057515034259, 4.77669608542845920936960296704, 5.68172656336782479474723788061, 6.51928663894432697277073329861, 7.69168087572007604559803206004, 9.512095852896473796445619418212, 10.65035641645083176573102091054, 11.47345213889979344343693410613, 12.58566046191863217902228732461, 13.58131506935203673400265372032
