| L(s) = 1 | + (−2.38 − 1.19i)2-s + (1.71 − 0.238i)3-s + (3.06 + 4.11i)4-s + (0.727 + 2.42i)5-s + (−4.37 − 1.48i)6-s + (0.202 + 0.469i)7-s + (−1.44 − 8.22i)8-s + (2.88 − 0.817i)9-s + (1.17 − 6.66i)10-s + (1.49 − 0.354i)11-s + (6.23 + 6.32i)12-s + (−4.71 − 3.09i)13-s + (0.0793 − 1.36i)14-s + (1.82 + 3.99i)15-s + (−3.45 + 11.5i)16-s + (−1.33 + 0.484i)17-s + ⋯ |
| L(s) = 1 | + (−1.68 − 0.847i)2-s + (0.990 − 0.137i)3-s + (1.53 + 2.05i)4-s + (0.325 + 1.08i)5-s + (−1.78 − 0.607i)6-s + (0.0764 + 0.177i)7-s + (−0.512 − 2.90i)8-s + (0.962 − 0.272i)9-s + (0.371 − 2.10i)10-s + (0.450 − 0.106i)11-s + (1.79 + 1.82i)12-s + (−1.30 − 0.859i)13-s + (0.0211 − 0.363i)14-s + (0.471 + 1.03i)15-s + (−0.862 + 2.88i)16-s + (−0.322 + 0.117i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.635427 - 0.134736i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.635427 - 0.134736i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.71 + 0.238i)T \) |
| good | 2 | \( 1 + (2.38 + 1.19i)T + (1.19 + 1.60i)T^{2} \) |
| 5 | \( 1 + (-0.727 - 2.42i)T + (-4.17 + 2.74i)T^{2} \) |
| 7 | \( 1 + (-0.202 - 0.469i)T + (-4.80 + 5.09i)T^{2} \) |
| 11 | \( 1 + (-1.49 + 0.354i)T + (9.82 - 4.93i)T^{2} \) |
| 13 | \( 1 + (4.71 + 3.09i)T + (5.14 + 11.9i)T^{2} \) |
| 17 | \( 1 + (1.33 - 0.484i)T + (13.0 - 10.9i)T^{2} \) |
| 19 | \( 1 + (-0.986 - 0.359i)T + (14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-0.103 + 0.239i)T + (-15.7 - 16.7i)T^{2} \) |
| 29 | \( 1 + (0.103 + 1.77i)T + (-28.8 + 3.36i)T^{2} \) |
| 31 | \( 1 + (5.13 - 0.599i)T + (30.1 - 7.14i)T^{2} \) |
| 37 | \( 1 + (5.04 + 4.23i)T + (6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (6.03 - 3.03i)T + (24.4 - 32.8i)T^{2} \) |
| 43 | \( 1 + (-4.19 - 4.44i)T + (-2.50 + 42.9i)T^{2} \) |
| 47 | \( 1 + (7.26 + 0.848i)T + (45.7 + 10.8i)T^{2} \) |
| 53 | \( 1 + (-4.74 + 8.21i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (4.82 + 1.14i)T + (52.7 + 26.4i)T^{2} \) |
| 61 | \( 1 + (7.70 - 10.3i)T + (-17.4 - 58.4i)T^{2} \) |
| 67 | \( 1 + (0.373 - 6.40i)T + (-66.5 - 7.77i)T^{2} \) |
| 71 | \( 1 + (0.896 - 5.08i)T + (-66.7 - 24.2i)T^{2} \) |
| 73 | \( 1 + (1.03 + 5.87i)T + (-68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-8.88 - 4.46i)T + (47.1 + 63.3i)T^{2} \) |
| 83 | \( 1 + (-11.5 - 5.81i)T + (49.5 + 66.5i)T^{2} \) |
| 89 | \( 1 + (1.71 + 9.71i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-1.03 + 3.44i)T + (-81.0 - 53.3i)T^{2} \) |
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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
−14.45824782597288216743762711666, −12.93474703720603036537414548147, −11.83453856866474517104425680103, −10.54416872078330315062874548976, −9.875789373439421969078153618594, −8.873487758739507447784824122875, −7.71204132256759362126205523326, −6.87117131731274078451387433775, −3.27278229191209588631189980987, −2.19511514743989928642532476910,
1.78469531244498435919842016086, 4.94988147862899352730689825929, 6.84795445479317731198182949351, 7.80711125731332191315600769670, 9.158984850424488719297315832303, 9.213650461005199371679324396116, 10.49517277142775370025351514899, 12.17489699228281643123589045754, 13.80364620890676650629480692807, 14.75741118556724189615965075556
