| L(s) = 1 | + (−1.74 + 1.26i)2-s + (−0.309 + 0.951i)3-s + (0.824 − 2.53i)4-s + (−0.460 + 2.18i)5-s + (−0.667 − 2.05i)6-s − 3.16·7-s + (0.446 + 1.37i)8-s + (−0.809 − 0.587i)9-s + (−1.97 − 4.40i)10-s + (−1.24 + 0.904i)11-s + (2.15 + 1.56i)12-s + (4.24 + 3.08i)13-s + (5.52 − 4.01i)14-s + (−1.93 − 1.11i)15-s + (1.79 + 1.30i)16-s + (0.398 + 1.22i)17-s + ⋯ |
| L(s) = 1 | + (−1.23 + 0.897i)2-s + (−0.178 + 0.549i)3-s + (0.412 − 1.26i)4-s + (−0.205 + 0.978i)5-s + (−0.272 − 0.838i)6-s − 1.19·7-s + (0.157 + 0.485i)8-s + (−0.269 − 0.195i)9-s + (−0.624 − 1.39i)10-s + (−0.375 + 0.272i)11-s + (0.623 + 0.452i)12-s + (1.17 + 0.854i)13-s + (1.47 − 1.07i)14-s + (−0.500 − 0.287i)15-s + (0.448 + 0.325i)16-s + (0.0967 + 0.297i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.958 - 0.284i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0560913 + 0.385636i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0560913 + 0.385636i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.309 - 0.951i)T \) |
| 5 | \( 1 + (0.460 - 2.18i)T \) |
| good | 2 | \( 1 + (1.74 - 1.26i)T + (0.618 - 1.90i)T^{2} \) |
| 7 | \( 1 + 3.16T + 7T^{2} \) |
| 11 | \( 1 + (1.24 - 0.904i)T + (3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-4.24 - 3.08i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-0.398 - 1.22i)T + (-13.7 + 9.99i)T^{2} \) |
| 19 | \( 1 + (-1.68 - 5.17i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + (-5.21 + 3.78i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.730 - 2.24i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.37 + 4.24i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-4.81 - 3.50i)T + (11.4 + 35.1i)T^{2} \) |
| 41 | \( 1 + (-6.90 - 5.01i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 8.48T + 43T^{2} \) |
| 47 | \( 1 + (-0.232 + 0.716i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-3.01 + 9.27i)T + (-42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (3.32 + 2.41i)T + (18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-8.65 + 6.28i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (0.586 + 1.80i)T + (-54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (0.0219 - 0.0674i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (3.24 - 2.35i)T + (22.5 - 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.500 + 1.53i)T + (-63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.06 - 12.5i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (5.88 - 4.27i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-3.15 + 9.69i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−15.38128047962581271437839293745, −14.50630526912874519006823040379, −12.95065221165124203351678744775, −11.25727409527901363571076969220, −10.20338619385923543048968551922, −9.479705750419797339312462464856, −8.197976221695140506942144336797, −6.82514288021290897793596326111, −6.10026616508541916535578510080, −3.56709129086432262265086044488,
0.806867749991927590267876103023, 3.10440129958088007915926726447, 5.62851780073715432403730556936, 7.42683906258453247164748114132, 8.667711463972035094798060232407, 9.406508738266010147883501390970, 10.72074032963159566798214424852, 11.68558941067040919521941651610, 12.86314498621735233997059498827, 13.37266330566050910827894233042
