| 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
−17.95491009566190088191155607466, −17.16011446073710286553063384639, −14.97622368255424904432113143096, −14.38309829842540245811524451835, −12.92713296327118282480992850514, −10.52014806856028509131170639158, −9.190405349251121533857899313835, −7.74495521778455492516873605838, −6.76638963313086023014787926262, −2.57798562571163687268247860353,
1.84200640226682905148493091293, 4.84409882738780592324317336628, 8.356362591043599945476078637608, 9.230312606116662960298990182488, 9.950603041491845854985965132423, 12.00996959749500378001330427347, 13.81758467740136338310575910444, 14.84011557429346394993484500218, 16.79034003953752222426035926216, 17.43548018981888451079900761166
