| L(s) = 1 | + (0.324 + 0.783i)2-s + (−1.35 + 0.906i)3-s + (2.32 − 2.32i)4-s + (−6.70 − 1.33i)5-s + (−1.14 − 0.768i)6-s + (0.886 − 0.176i)7-s + (5.70 + 2.36i)8-s + (−2.42 + 5.85i)9-s + (−1.13 − 5.68i)10-s + (3.73 − 5.59i)11-s + (−1.04 + 5.24i)12-s + (10.5 + 10.5i)13-s + (0.425 + 0.637i)14-s + (10.2 − 4.26i)15-s − 7.89i·16-s + (−14.7 + 8.37i)17-s + ⋯ |
| L(s) = 1 | + (0.162 + 0.391i)2-s + (−0.451 + 0.302i)3-s + (0.580 − 0.580i)4-s + (−1.34 − 0.266i)5-s + (−0.191 − 0.128i)6-s + (0.126 − 0.0251i)7-s + (0.712 + 0.295i)8-s + (−0.269 + 0.650i)9-s + (−0.113 − 0.568i)10-s + (0.339 − 0.508i)11-s + (−0.0869 + 0.437i)12-s + (0.812 + 0.812i)13-s + (0.0304 + 0.0455i)14-s + (0.686 − 0.284i)15-s − 0.493i·16-s + (−0.870 + 0.492i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.934 - 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.766793 + 0.140779i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.766793 + 0.140779i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + (14.7 - 8.37i)T \) |
| good | 2 | \( 1 + (-0.324 - 0.783i)T + (-2.82 + 2.82i)T^{2} \) |
| 3 | \( 1 + (1.35 - 0.906i)T + (3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (6.70 + 1.33i)T + (23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (-0.886 + 0.176i)T + (45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-3.73 + 5.59i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (-10.5 - 10.5i)T + 169iT^{2} \) |
| 19 | \( 1 + (12.9 + 31.3i)T + (-255. + 255. i)T^{2} \) |
| 23 | \( 1 + (-15.4 - 10.3i)T + (202. + 488. i)T^{2} \) |
| 29 | \( 1 + (-4.13 + 20.7i)T + (-776. - 321. i)T^{2} \) |
| 31 | \( 1 + (-21.1 - 31.6i)T + (-367. + 887. i)T^{2} \) |
| 37 | \( 1 + (33.3 - 22.2i)T + (523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (3.70 - 0.736i)T + (1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (5.21 - 12.5i)T + (-1.30e3 - 1.30e3i)T^{2} \) |
| 47 | \( 1 + (9.20 + 9.20i)T + 2.20e3iT^{2} \) |
| 53 | \( 1 + (-0.763 - 1.84i)T + (-1.98e3 + 1.98e3i)T^{2} \) |
| 59 | \( 1 + (-31.7 - 13.1i)T + (2.46e3 + 2.46e3i)T^{2} \) |
| 61 | \( 1 + (-6.92 - 34.8i)T + (-3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 + 31.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (-86.1 + 57.5i)T + (1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (-61.7 - 12.2i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (36.2 - 54.3i)T + (-2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (39.7 - 16.4i)T + (4.87e3 - 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-49.5 + 49.5i)T - 7.92e3iT^{2} \) |
| 97 | \( 1 + (-24.9 + 125. i)T + (-8.69e3 - 3.60e3i)T^{2} \) |
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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
−19.30011849514729347077958816771, −17.10553802449320994120740421434, −15.97686325023402682125620682325, −15.34354848889165274570572185633, −13.67616567414434418914938713950, −11.49701760429148642502629807856, −10.96964027808346680313878987777, −8.471639006345098481783554042851, −6.66682316665691600129025257471, −4.66648495643853188972415367283,
3.72195384313266909951717340086, 6.73993307604006132933322756889, 8.196592445453276961473738077109, 10.87410092471400360567609839881, 11.83829318999568236243454009309, 12.73206349338761435725187856287, 14.94548161313213022496552183491, 16.03497238197434444901296038374, 17.39689110847739477535291531988, 18.77945846983229720570913882584
