| 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
−14.75741118556724189615965075556, −13.80364620890676650629480692807, −12.17489699228281643123589045754, −10.49517277142775370025351514899, −9.213650461005199371679324396116, −9.158984850424488719297315832303, −7.80711125731332191315600769670, −6.84795445479317731198182949351, −4.94988147862899352730689825929, −1.78469531244498435919842016086,
2.19511514743989928642532476910, 3.27278229191209588631189980987, 6.87117131731274078451387433775, 7.71204132256759362126205523326, 8.873487758739507447784824122875, 9.875789373439421969078153618594, 10.54416872078330315062874548976, 11.83453856866474517104425680103, 12.93474703720603036537414548147, 14.45824782597288216743762711666
