L(s) = 1 | + (1.20 + 0.741i)2-s + (0.899 + 1.78i)4-s + (0.431 + 0.431i)5-s − 0.145i·7-s + (−0.242 + 2.81i)8-s + (0.199 + 0.839i)10-s + (2.42 + 2.42i)11-s + (0.201 − 0.201i)13-s + (0.107 − 0.174i)14-s + (−2.38 + 3.21i)16-s − 3.15·17-s + (3.09 − 3.09i)19-s + (−0.382 + 1.15i)20-s + (1.12 + 4.72i)22-s + 5.99i·23-s + ⋯ |
L(s) = 1 | + (0.851 + 0.524i)2-s + (0.449 + 0.893i)4-s + (0.192 + 0.192i)5-s − 0.0548i·7-s + (−0.0856 + 0.996i)8-s + (0.0630 + 0.265i)10-s + (0.731 + 0.731i)11-s + (0.0558 − 0.0558i)13-s + (0.0287 − 0.0466i)14-s + (−0.595 + 0.803i)16-s − 0.764·17-s + (0.709 − 0.709i)19-s + (−0.0855 + 0.259i)20-s + (0.239 + 1.00i)22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82559 + 1.42490i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82559 + 1.42490i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.20 - 0.741i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-0.431 - 0.431i)T + 5iT^{2} \) |
| 7 | \( 1 + 0.145iT - 7T^{2} \) |
| 11 | \( 1 + (-2.42 - 2.42i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.201 + 0.201i)T - 13iT^{2} \) |
| 17 | \( 1 + 3.15T + 17T^{2} \) |
| 19 | \( 1 + (-3.09 + 3.09i)T - 19iT^{2} \) |
| 23 | \( 1 - 5.99iT - 23T^{2} \) |
| 29 | \( 1 + (-0.395 + 0.395i)T - 29iT^{2} \) |
| 31 | \( 1 - 3.98T + 31T^{2} \) |
| 37 | \( 1 + (5.29 + 5.29i)T + 37iT^{2} \) |
| 41 | \( 1 + 1.76iT - 41T^{2} \) |
| 43 | \( 1 + (8.07 + 8.07i)T + 43iT^{2} \) |
| 47 | \( 1 - 5.91T + 47T^{2} \) |
| 53 | \( 1 + (7.91 + 7.91i)T + 53iT^{2} \) |
| 59 | \( 1 + (-6.18 - 6.18i)T + 59iT^{2} \) |
| 61 | \( 1 + (0.789 - 0.789i)T - 61iT^{2} \) |
| 67 | \( 1 + (-6.34 + 6.34i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.39iT - 71T^{2} \) |
| 73 | \( 1 + 2.23iT - 73T^{2} \) |
| 79 | \( 1 + 7.01T + 79T^{2} \) |
| 83 | \( 1 + (4.08 - 4.08i)T - 83iT^{2} \) |
| 89 | \( 1 + 14.1iT - 89T^{2} \) |
| 97 | \( 1 - 17.2T + 97T^{2} \) |
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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
−11.66118744327063725120806781177, −10.59042034295977478095866845479, −9.438729303681091355791495766745, −8.500095906354873210197747797022, −7.27895095562445173166460339824, −6.73027439498657051640040305076, −5.60689241861021485827467475025, −4.59038415132465386554230150130, −3.54760547806585710475203766366, −2.15394303123347388833483022333,
1.33895597566546029253695944448, 2.87395339910692530934052611055, 3.99870647054185992194545375203, 5.06744583477059972876239343323, 6.10699151835090419695343472530, 6.87423294517590988267576877491, 8.359918773751252855960801287210, 9.333488636642792495666148489576, 10.23891269320445948944860524428, 11.17505093812302476684972379847