| L(s) = 1 | + (−3.73 + 0.658i)2-s + (0.577 + 0.687i)3-s + (−1.50 + 0.547i)4-s + (−39.5 − 14.3i)5-s + (−2.61 − 2.19i)6-s + (−34.7 + 60.1i)7-s + (57.8 − 33.3i)8-s + (13.9 − 78.9i)9-s + (157. + 27.7i)10-s + (51.3 + 88.9i)11-s + (−1.24 − 0.719i)12-s + (−90.5 + 107. i)13-s + (90.0 − 247. i)14-s + (−12.9 − 35.4i)15-s + (−174. + 146. i)16-s + (−34.7 − 197. i)17-s + ⋯ |
| L(s) = 1 | + (−0.934 + 0.164i)2-s + (0.0641 + 0.0764i)3-s + (−0.0940 + 0.0342i)4-s + (−1.58 − 0.575i)5-s + (−0.0725 − 0.0608i)6-s + (−0.708 + 1.22i)7-s + (0.903 − 0.521i)8-s + (0.171 − 0.975i)9-s + (1.57 + 0.277i)10-s + (0.424 + 0.734i)11-s + (−0.00864 − 0.00499i)12-s + (−0.535 + 0.638i)13-s + (0.459 − 1.26i)14-s + (−0.0574 − 0.157i)15-s + (−0.681 + 0.572i)16-s + (−0.120 − 0.682i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00766i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.000303965 - 0.0793032i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.000303965 - 0.0793032i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (354. + 68.1i)T \) |
| good | 2 | \( 1 + (3.73 - 0.658i)T + (15.0 - 5.47i)T^{2} \) |
| 3 | \( 1 + (-0.577 - 0.687i)T + (-14.0 + 79.7i)T^{2} \) |
| 5 | \( 1 + (39.5 + 14.3i)T + (478. + 401. i)T^{2} \) |
| 7 | \( 1 + (34.7 - 60.1i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-51.3 - 88.9i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (90.5 - 107. i)T + (-4.95e3 - 2.81e4i)T^{2} \) |
| 17 | \( 1 + (34.7 + 197. i)T + (-7.84e4 + 2.85e4i)T^{2} \) |
| 23 | \( 1 + (173. - 63.1i)T + (2.14e5 - 1.79e5i)T^{2} \) |
| 29 | \( 1 + (1.00e3 + 176. i)T + (6.64e5 + 2.41e5i)T^{2} \) |
| 31 | \( 1 + (-12.3 - 7.15i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 1.15e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (1.46e3 + 1.74e3i)T + (-4.90e5 + 2.78e6i)T^{2} \) |
| 43 | \( 1 + (-1.82e3 - 662. i)T + (2.61e6 + 2.19e6i)T^{2} \) |
| 47 | \( 1 + (-72.7 + 412. i)T + (-4.58e6 - 1.66e6i)T^{2} \) |
| 53 | \( 1 + (-1.30e3 - 3.57e3i)T + (-6.04e6 + 5.07e6i)T^{2} \) |
| 59 | \( 1 + (100. - 17.6i)T + (1.13e7 - 4.14e6i)T^{2} \) |
| 61 | \( 1 + (-2.05e3 + 746. i)T + (1.06e7 - 8.89e6i)T^{2} \) |
| 67 | \( 1 + (-839. - 148. i)T + (1.89e7 + 6.89e6i)T^{2} \) |
| 71 | \( 1 + (253. - 697. i)T + (-1.94e7 - 1.63e7i)T^{2} \) |
| 73 | \( 1 + (3.82e3 - 3.21e3i)T + (4.93e6 - 2.79e7i)T^{2} \) |
| 79 | \( 1 + (5.79e3 + 6.90e3i)T + (-6.76e6 + 3.83e7i)T^{2} \) |
| 83 | \( 1 + (2.76e3 - 4.79e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (3.86e3 - 4.60e3i)T + (-1.08e7 - 6.17e7i)T^{2} \) |
| 97 | \( 1 + (-2.95e3 + 521. i)T + (8.31e7 - 3.02e7i)T^{2} \) |
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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
−18.58777498931659047165661399875, −17.10469112147559319470907495781, −15.89254250414878820449500428309, −15.08546109539475568898035080863, −12.63803937773561470603028909480, −11.83382382290380109952784767758, −9.489636912063680369269663857370, −8.716197274827216129884725679739, −7.11549361941374260813933115517, −4.18898855961283458123643478990,
0.096921788917534565244022354767, 3.99028999611986954585158226932, 7.29380861413308829894124385783, 8.269459695360343753280664202705, 10.31417089703630356656456908795, 11.08052651695158014637747785926, 13.09236237624853270345003063070, 14.56238373038877533839381208263, 16.20632659028251477400182768597, 17.05333058872321315826106782427
