| 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
−13.37266330566050910827894233042, −12.86314498621735233997059498827, −11.68558941067040919521941651610, −10.72074032963159566798214424852, −9.406508738266010147883501390970, −8.667711463972035094798060232407, −7.42683906258453247164748114132, −5.62851780073715432403730556936, −3.10440129958088007915926726447, −0.806867749991927590267876103023,
3.56709129086432262265086044488, 6.10026616508541916535578510080, 6.82514288021290897793596326111, 8.197976221695140506942144336797, 9.479705750419797339312462464856, 10.20338619385923543048968551922, 11.25727409527901363571076969220, 12.95065221165124203351678744775, 14.50630526912874519006823040379, 15.38128047962581271437839293745
