| L(s) = 1 | + (0.326 − 1.85i)2-s + (−0.766 − 0.642i)3-s + (−1.43 − 0.524i)4-s + (−0.826 + 0.300i)5-s + (−1.43 + 1.20i)6-s + (1.43 + 2.49i)7-s + (0.439 − 0.761i)8-s + (0.173 + 0.984i)9-s + (0.286 + 1.62i)10-s + (0.918 − 1.59i)11-s + (0.766 + 1.32i)12-s + (−2.11 + 1.77i)13-s + (5.08 − 1.85i)14-s + (0.826 + 0.300i)15-s + (−3.61 − 3.03i)16-s + (−1.23 + 6.99i)17-s + ⋯ |
| L(s) = 1 | + (0.230 − 1.30i)2-s + (−0.442 − 0.371i)3-s + (−0.719 − 0.262i)4-s + (−0.369 + 0.134i)5-s + (−0.587 + 0.493i)6-s + (0.544 + 0.942i)7-s + (0.155 − 0.269i)8-s + (0.0578 + 0.328i)9-s + (0.0907 + 0.514i)10-s + (0.277 − 0.479i)11-s + (0.221 + 0.383i)12-s + (−0.586 + 0.491i)13-s + (1.35 − 0.494i)14-s + (0.213 + 0.0776i)15-s + (−0.903 − 0.757i)16-s + (−0.299 + 1.69i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0970 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0970 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.580210 - 0.639562i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580210 - 0.639562i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.766 + 0.642i)T \) |
| 19 | \( 1 + (-3.93 + 1.86i)T \) |
| good | 2 | \( 1 + (-0.326 + 1.85i)T + (-1.87 - 0.684i)T^{2} \) |
| 5 | \( 1 + (0.826 - 0.300i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.43 - 2.49i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.918 + 1.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.11 - 1.77i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (1.23 - 6.99i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (6.19 + 2.25i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.543 + 3.08i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (3.82 + 6.62i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 + (-3.05 - 2.56i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-10.7 + 3.91i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.383 - 2.17i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (2.53 + 0.924i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (1.46 - 8.28i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (0.578 + 0.210i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.638 + 3.61i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.00 + 2.54i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (7.66 + 6.43i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-1.23 - 1.03i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.492 + 0.853i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-13.0 + 10.9i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.02 - 5.81i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−14.76410687475959123615892895820, −13.43255996300306597721335435832, −12.26891564207712002573255339013, −11.67029051175893974544806279877, −10.80709515414666387171491721383, −9.351251832260926352776651176113, −7.76273569442966573181052283566, −5.94540037199828716197155307548, −4.11590931235712046424992216906, −2.12470280408781661591321108240,
4.34511705558270003881714561954, 5.44549826801597064547053299234, 7.10200700693017552283587643322, 7.81807022465561222102096894148, 9.591177916359654516006709743684, 10.99732548433986611572634648686, 12.14755077989390679361025427544, 13.88645216570967729365082955251, 14.48274740407572000783809724054, 15.86882622858447857514556964062
