| L(s) = 1 | + (−0.881 + 2.42i)2-s + (−0.384 + 0.0678i)3-s + (−2.02 − 1.69i)4-s + (1.73 − 1.45i)5-s + (0.174 − 0.991i)6-s + (5.72 − 9.91i)7-s + (−3.03 + 1.74i)8-s + (−8.31 + 3.02i)9-s + (1.99 + 5.47i)10-s + (3.88 + 6.73i)11-s + (0.893 + 0.515i)12-s + (−13.5 − 2.39i)13-s + (18.9 + 22.5i)14-s + (−0.567 + 0.676i)15-s + (−3.40 − 19.2i)16-s + (2.79 + 1.01i)17-s + ⋯ |
| L(s) = 1 | + (−0.440 + 1.21i)2-s + (−0.128 + 0.0226i)3-s + (−0.506 − 0.424i)4-s + (0.346 − 0.290i)5-s + (0.0291 − 0.165i)6-s + (0.817 − 1.41i)7-s + (−0.378 + 0.218i)8-s + (−0.923 + 0.336i)9-s + (0.199 + 0.547i)10-s + (0.353 + 0.611i)11-s + (0.0744 + 0.0429i)12-s + (−1.04 − 0.184i)13-s + (1.35 + 1.61i)14-s + (−0.0378 + 0.0450i)15-s + (−0.212 − 1.20i)16-s + (0.164 + 0.0598i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.304 - 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.577737 + 0.421930i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.577737 + 0.421930i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-17.7 - 6.75i)T \) |
| good | 2 | \( 1 + (0.881 - 2.42i)T + (-3.06 - 2.57i)T^{2} \) |
| 3 | \( 1 + (0.384 - 0.0678i)T + (8.45 - 3.07i)T^{2} \) |
| 5 | \( 1 + (-1.73 + 1.45i)T + (4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (-5.72 + 9.91i)T + (-24.5 - 42.4i)T^{2} \) |
| 11 | \( 1 + (-3.88 - 6.73i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (13.5 + 2.39i)T + (158. + 57.8i)T^{2} \) |
| 17 | \( 1 + (-2.79 - 1.01i)T + (221. + 185. i)T^{2} \) |
| 23 | \( 1 + (20.9 + 17.5i)T + (91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-6.50 - 17.8i)T + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-28.8 - 16.6i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + 2.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (52.5 - 9.27i)T + (1.57e3 - 574. i)T^{2} \) |
| 43 | \( 1 + (3.40 - 2.85i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (-20.0 + 7.29i)T + (1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 + (-47.6 + 56.7i)T + (-487. - 2.76e3i)T^{2} \) |
| 59 | \( 1 + (-18.5 + 51.0i)T + (-2.66e3 - 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-81.0 - 68.0i)T + (646. + 3.66e3i)T^{2} \) |
| 67 | \( 1 + (10.5 + 29.0i)T + (-3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (49.1 + 58.6i)T + (-875. + 4.96e3i)T^{2} \) |
| 73 | \( 1 + (-3.15 - 17.9i)T + (-5.00e3 + 1.82e3i)T^{2} \) |
| 79 | \( 1 + (30.0 - 5.29i)T + (5.86e3 - 2.13e3i)T^{2} \) |
| 83 | \( 1 + (41.2 - 71.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (86.5 + 15.2i)T + (7.44e3 + 2.70e3i)T^{2} \) |
| 97 | \( 1 + (18.6 - 51.1i)T + (-7.20e3 - 6.04e3i)T^{2} \) |
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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
−17.84538547371175103461877322624, −17.18128716738504666063046766988, −16.46774842150073143035911867440, −14.71882135748665475313979847645, −13.94569984675427976476498200022, −11.82725412371050334011887853803, −10.04331161720345781288186412535, −8.247669903156382441925502833465, −7.08854562074849904480263488623, −5.15118769448373011118286068546,
2.56909808272184381686094111302, 5.81421001133592366861712499621, 8.601194436998005701564244988257, 9.841378094279541434986892260065, 11.66492266170330471996043893599, 11.88899521184661963448516632249, 14.10266860273773270055463030968, 15.37513442608707079124651351249, 17.41607842781735298774732756482, 18.26592028283101535333407589619
