| L(s) = 1 | + (−3.97 − 2.29i)2-s + (10.1 + 5.84i)3-s + (2.50 + 4.34i)4-s + (20.8 − 36.0i)5-s + (−26.8 − 46.4i)6-s + 24.1·7-s + 50.3i·8-s + (27.9 + 48.3i)9-s + (−165. + 95.5i)10-s + 62.4·11-s + 58.6i·12-s + (−250. + 144. i)13-s + (−96.0 − 55.4i)14-s + (422. − 243. i)15-s + (155. − 269. i)16-s + (−12.2 + 21.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.992 − 0.573i)2-s + (1.12 + 0.649i)3-s + (0.156 + 0.271i)4-s + (0.833 − 1.44i)5-s + (−0.744 − 1.29i)6-s + 0.493·7-s + 0.786i·8-s + (0.344 + 0.597i)9-s + (−1.65 + 0.955i)10-s + 0.516·11-s + 0.407i·12-s + (−1.48 + 0.855i)13-s + (−0.489 − 0.282i)14-s + (1.87 − 1.08i)15-s + (0.607 − 1.05i)16-s + (−0.0424 + 0.0734i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.693 + 0.720i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.03909 - 0.441756i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03909 - 0.441756i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (283. - 223. i)T \) |
| good | 2 | \( 1 + (3.97 + 2.29i)T + (8 + 13.8i)T^{2} \) |
| 3 | \( 1 + (-10.1 - 5.84i)T + (40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (-20.8 + 36.0i)T + (-312.5 - 541. i)T^{2} \) |
| 7 | \( 1 - 24.1T + 2.40e3T^{2} \) |
| 11 | \( 1 - 62.4T + 1.46e4T^{2} \) |
| 13 | \( 1 + (250. - 144. i)T + (1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 + (12.2 - 21.2i)T + (-4.17e4 - 7.23e4i)T^{2} \) |
| 23 | \( 1 + (-302. - 524. i)T + (-1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-249. + 144. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 - 418. iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.02e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (717. + 414. i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (-632. + 1.09e3i)T + (-1.70e6 - 2.96e6i)T^{2} \) |
| 47 | \( 1 + (421. + 729. i)T + (-2.43e6 + 4.22e6i)T^{2} \) |
| 53 | \( 1 + (-3.14e3 + 1.81e3i)T + (3.94e6 - 6.83e6i)T^{2} \) |
| 59 | \( 1 + (4.62e3 + 2.67e3i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-1.47e3 - 2.54e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-4.19e3 + 2.42e3i)T + (1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 + (4.78e3 + 2.76e3i)T + (1.27e7 + 2.20e7i)T^{2} \) |
| 73 | \( 1 + (-792. + 1.37e3i)T + (-1.41e7 - 2.45e7i)T^{2} \) |
| 79 | \( 1 + (-4.62e3 - 2.67e3i)T + (1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + 740.T + 4.74e7T^{2} \) |
| 89 | \( 1 + (-4.78e3 + 2.76e3i)T + (3.13e7 - 5.43e7i)T^{2} \) |
| 97 | \( 1 + (-2.96e3 - 1.71e3i)T + (4.42e7 + 7.66e7i)T^{2} \) |
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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
−17.43548018981888451079900761166, −16.79034003953752222426035926216, −14.84011557429346394993484500218, −13.81758467740136338310575910444, −12.00996959749500378001330427347, −9.950603041491845854985965132423, −9.230312606116662960298990182488, −8.356362591043599945476078637608, −4.84409882738780592324317336628, −1.84200640226682905148493091293,
2.57798562571163687268247860353, 6.76638963313086023014787926262, 7.74495521778455492516873605838, 9.190405349251121533857899313835, 10.52014806856028509131170639158, 12.92713296327118282480992850514, 14.38309829842540245811524451835, 14.97622368255424904432113143096, 17.16011446073710286553063384639, 17.95491009566190088191155607466
