| L(s) = 1 | + (0.984 + 0.173i)2-s + (−1.54 − 0.775i)3-s + (0.939 + 0.342i)4-s + (1.21 + 1.87i)5-s + (−1.39 − 1.03i)6-s + (−1.48 + 0.856i)7-s + (0.866 + 0.5i)8-s + (1.79 + 2.40i)9-s + (0.873 + 2.05i)10-s + (−0.232 − 0.134i)11-s + (−1.19 − 1.25i)12-s + (−2.84 + 2.38i)13-s + (−1.61 + 0.586i)14-s + (−0.431 − 3.84i)15-s + (0.766 + 0.642i)16-s + (−0.267 + 1.51i)17-s + ⋯ |
| L(s) = 1 | + (0.696 + 0.122i)2-s + (−0.894 − 0.447i)3-s + (0.469 + 0.171i)4-s + (0.544 + 0.838i)5-s + (−0.567 − 0.421i)6-s + (−0.560 + 0.323i)7-s + (0.306 + 0.176i)8-s + (0.599 + 0.800i)9-s + (0.276 + 0.650i)10-s + (−0.0702 − 0.0405i)11-s + (−0.343 − 0.363i)12-s + (−0.787 + 0.661i)13-s + (−0.430 + 0.156i)14-s + (−0.111 − 0.993i)15-s + (0.191 + 0.160i)16-s + (−0.0648 + 0.367i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19961 + 0.973425i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19961 + 0.973425i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.54 + 0.775i)T \) |
| 5 | \( 1 + (-1.21 - 1.87i)T \) |
| 19 | \( 1 + (-2.37 - 3.65i)T \) |
| good | 7 | \( 1 + (1.48 - 0.856i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (0.232 + 0.134i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (2.84 - 2.38i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.267 - 1.51i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-0.801 - 0.291i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.356 - 2.02i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-0.244 + 0.141i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 9.38T + 37T^{2} \) |
| 41 | \( 1 + (-1.94 - 1.62i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.01 - 5.53i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.42 + 8.10i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (1.09 - 2.99i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-2.44 + 13.8i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (1.11 + 0.404i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.393 + 2.23i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.24 - 0.817i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (4.67 - 5.57i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-3.38 + 4.03i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.43 + 12.8i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-10.6 + 8.94i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.04 + 5.92i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−11.14423054344273526107725833361, −10.19051626752663500588113280401, −9.492713274228082194005576978950, −7.85252344509802016546277167397, −7.01592142216483658258130469678, −6.25994676050043961129732306693, −5.64862398245014152337970020925, −4.51869711640124983668111691037, −3.08455969437606637805592850910, −1.89414200508424652284630820682,
0.78839965382822109277722893989, 2.71990711731025913458110910509, 4.15619207590957215359138318388, 4.99046152867594089745649629375, 5.68954251721810128257621608474, 6.62614091519669529180530068027, 7.63199814927172584089625181844, 9.180787868278473708101795133020, 9.809795346483819042676685221106, 10.56063558404791006714685308735
