| L(s) = 1 | + (0.675 − 1.63i)2-s + (2.24 − 3.36i)3-s + (0.624 + 0.624i)4-s + (−1.50 − 7.58i)5-s + (−3.96 − 5.94i)6-s + (−0.798 + 4.01i)7-s + (7.96 − 3.29i)8-s + (−2.82 − 6.81i)9-s + (−13.3 − 2.66i)10-s + (5.77 − 3.85i)11-s + (3.50 − 0.697i)12-s + (−2.37 + 2.37i)13-s + (6.00 + 4.01i)14-s + (−28.9 − 11.9i)15-s − 11.6i·16-s + ⋯ |
| L(s) = 1 | + (0.337 − 0.815i)2-s + (0.749 − 1.12i)3-s + (0.156 + 0.156i)4-s + (−0.301 − 1.51i)5-s + (−0.661 − 0.990i)6-s + (−0.114 + 0.573i)7-s + (0.995 − 0.412i)8-s + (−0.313 − 0.757i)9-s + (−1.33 − 0.266i)10-s + (0.524 − 0.350i)11-s + (0.292 − 0.0581i)12-s + (−0.182 + 0.182i)13-s + (0.429 + 0.286i)14-s + (−1.92 − 0.798i)15-s − 0.730i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.759164 - 2.51340i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.759164 - 2.51340i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.675 + 1.63i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-2.24 + 3.36i)T + (-3.44 - 8.31i)T^{2} \) |
| 5 | \( 1 + (1.50 + 7.58i)T + (-23.0 + 9.56i)T^{2} \) |
| 7 | \( 1 + (0.798 - 4.01i)T + (-45.2 - 18.7i)T^{2} \) |
| 11 | \( 1 + (-5.77 + 3.85i)T + (46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (2.37 - 2.37i)T - 169iT^{2} \) |
| 19 | \( 1 + (9.43 - 22.7i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (6.40 + 9.57i)T + (-202. + 488. i)T^{2} \) |
| 29 | \( 1 + (11.7 - 2.34i)T + (776. - 321. i)T^{2} \) |
| 31 | \( 1 + (-31.4 - 21.0i)T + (367. + 887. i)T^{2} \) |
| 37 | \( 1 + (17.9 - 26.8i)T + (-523. - 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-6.59 + 33.1i)T + (-1.55e3 - 643. i)T^{2} \) |
| 43 | \( 1 + (1.33 + 3.21i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (3.16 - 3.16i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-11.7 + 28.3i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (10.8 - 4.49i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (69.5 + 13.8i)T + (3.43e3 + 1.42e3i)T^{2} \) |
| 67 | \( 1 - 28.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (19.0 - 28.5i)T + (-1.92e3 - 4.65e3i)T^{2} \) |
| 73 | \( 1 + (18.2 + 91.8i)T + (-4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-17.8 + 11.9i)T + (2.38e3 - 5.76e3i)T^{2} \) |
| 83 | \( 1 + (-104. - 43.4i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-28.3 - 28.3i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-163. + 32.4i)T + (8.69e3 - 3.60e3i)T^{2} \) |
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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
−11.87452777520900388338561778793, −10.37063400910830172690737274081, −9.003674804116539387574278442494, −8.373636366586368320443304490477, −7.57072073883588104951648418014, −6.29316252466296392201575314965, −4.76149960129892837470652526609, −3.56474282872491304244977751830, −2.16942667826027409705461822153, −1.20360135709817727892337650454,
2.55738674976430281540603391552, 3.76638080079863834511756562090, 4.66751828134849454696226224133, 6.23267716947092255610521920441, 7.04170497714283433610718151044, 7.82002503790251170489807030869, 9.227333450440371561240963115248, 10.23201918587462094502299413969, 10.72114596640790940314520212037, 11.63532345818701317195362765995
